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Table of Contents
Precalculus
Limits
2.5 Formal Definition of the Limit
2.6 Proofs of Some Basic Limit Rules
Differentiation
Basics of Differentiation
3.2 Product and Quotient Rules
3.3 Derivatives of Trigonometric Functions
3.5 Higher Order Derivatives - An introduction to second order derivatives
3.7 Derivatives of Exponential and Logarithm Functions
Applications of Derivatives
3.11 Extrema and Points of Inflection
Integration
Basics of Integration
4.2 Fundamental Theorem of Calculus
Integration Techniques
4.6 Derivative Rules and the Substitution Rule
4.8 Trigonometric Substitutions
4.10 Rational Functions by Partial Fraction Decomposition
4.11 Tangent Half Angle Substitution
Applications of Integration
4.18 Volume of solids of revolution
Parametric and Polar Equations
Parametric Equations
- Introduction to Parametric Equations
- Differentiation and Parametric Equations
- Integration and Parametric Equations
- Exercises
Polar Equations
Sequences and Series
Basics
Series and calculus
Multivariable and Differential Calculus
Extensions
Advanced Integration Techniques
Further Analysis
Formal Theory of Calculus
Appendix
- Choosing delta
Solutions
- Precalculus/Solutions
- Infinity is not a number/Solutions
- Limits/Solutions
- Differentiation/Differentiation Defined/Solutions
- Chain Rule/Solutions
- Some Important Theorems/Solutions
- Differentiation/Basics of Differentiation/Solutions
- L'Hôpital's rule/Solutions
- Related Rates/Solutions
- Differentiation/Applications of Derivatives/Solutions
- Integration/Solutions
References
Acknowledgements and Further Reading
Introduction
What is calculus?
Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. As a typical application of the methods of calculus, consider a moving car. It is possible to create a function describing the displacement of the car (where it is located in relation to a reference point) at any point in time as well as a function describing the velocity (speed and direction of movement) of the car at any point in time. If the car were traveling at a constant velocity, then algebra would be sufficient to determine the position of the car at any time; if the velocity is unknown but still constant, the position of the car could be used (along with the time) to find the velocity.
However, the velocity of a car cannot jump from zero to 35 miles per hour at the beginning of a trip, stay constant throughout, and then jump back to zero at the end. As the accelerator is pressed down, the velocity rises gradually, and usually not at a constant rate (i.e., the driver may push on the gas pedal harder at the beginning, in order to speed up). Describing such motion and finding velocities and distances at particular times cannot be done using methods taught in pre-calculus, whereas it is not only possible but straightforward with calculus.
Calculus has two basic applications: differential calculus and integral calculus. The simplest introduction to differential calculus involves an explicit series of numbers. Given the series (42, 43, 3, 18, 34), the differential of this series would be (1, -40, 15, 16). The new series is derived from the difference of successive numbers which gives rise to its name "differential". Rarely, if ever, are differentials used on an explicit series of numbers as done here. Instead, they are derived from a continuous function in a manner which is described later.
Integral calculus, like differential calculus, can also be introduced via series of numbers. Notice that in the previous example, the original series can almost be derived solely from its differential. Instead of taking the difference, however, integration involves taking the sum. Given the first number of the original series, 42 in this case, the rest of the original series can be derived by adding each successive number in its differential (42+1, 43-40, 3+15, 18+16). Note that knowledge of the first number in the original series is crucial in deriving the integral. As with differentials, integration is performed on continuous functions rather than explicit series of numbers, but the concept is still the same. Integral calculus allows us to calculate the area under a curve of almost any shape; in the car example, this enables you to find the displacement of the car based on the velocity curve. This is because the area under the curve is the total distance moved, as we will soon see. Let's understand this section very carefully. Suppose we have to add the numbers in series which is continuously "on" like 23,25,24,25,34,45,46,47, and so on...at this type integral calculation is very useful instead of the typical mathematical formulas.
Why learn calculus?
Calculus is essential for many areas of science and engineering. Both make heavy use of mathematical functions to describe and predict physical phenomena that are subject to continual change, and this requires the use of calculus. Take our car example: if you want to design cars, you need to know how to calculate forces, velocities, accelerations, and positions. All require calculus. Calculus is also necessary to study the motion of gases and particles, the interaction of forces, and the transfer of energy. It is also useful in business whenever rates are involved. For example, equations involving interest or supply and demand curves are grounded in the language of calculus.
Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry.
Notwithstanding calculus' functional utility (pun intended), many non-scientists and non-engineers have chosen to study calculus just for the challenge of doing so. A smaller number of persons undertake such a challenge and then discover that calculus is beautiful in and of itself.
What is involved in learning calculus?
Learning calculus, like much of mathematics, involves two parts:
- Understanding the concepts: You must be able to explain what it means when you take a derivative rather than merely apply the formulas for finding a derivative. Otherwise, you will have no idea whether or not your solution is correct. Drawing diagrams, for example, can help clarify abstract concepts.
- Symbolic manipulation: Like other branches of mathematics, calculus is written in symbols that represent concepts. You will learn what these symbols mean and how to use them. A good working knowledge of trigonometry and algebra is a must, especially in integral calculus. Sometimes you will need to manipulate expressions into a usable form before it is possible to perform operations in calculus.
What you should know before using this text
There are some basic skills that you need before you can use this text. Continuing with our example of a moving car:
- You will need to describe the motion of the car in symbols. This involves understanding functions.
- You need to manipulate these functions. This involves algebra.
- You need to translate symbols into graphs and vice-versa. This involves understanding the graphing of functions.
- It also helps (although it isn't necessarily essential) if you understand the functions used in trigonometry since these functions appear frequently in science.
Scope
The first four chapters of this textbook cover the topics taught in a typical high school or first year college course. The first chapter, Precalculus, reviews those aspects of functions most essential to the mastery of calculus. The second, Limits, introduces the concept of the limit process. It also discusses some applications of limits and proposes using limits to examine slope and area of functions. The next two chapters, Differentiation and Integration, apply limits to calculate derivatives and integrals. The Fundamental Theorem of Calculus is used, as are the essential formulae for computation of derivatives and integrals without resorting to the limit process. The third and fourth chapters include articles that apply the concepts previously learned to calculating volumes, and so on as well as other important formulae.
The remainder of the central Calculus chapters cover topics taught in higher-level calculus topics: multivariable calculus, vectors, and series (Taylor, convergent, divergent).
Finally, the other chapters cover the same material, using formal notation. They introduce the material at a much faster pace, and cover many more theorems than the other two sections. They assume knowledge of some set theory and set notation.
Precalculus
<h1> 1.1 Algebra</h1>
This section is intended to review algebraic manipulation. It is important to understand algebra in order to do calculus. If you have a good knowledge of algebra, you should probably just skim this section to be sure you are familiar with the ideas.
Rules of arithmetic and algebra
The following laws are true for all a, b, and c, whether a, b, and c are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.
Addition
- Commutative Law:
.
- Associative Law:
.
- Additive Identity:
.
- Additive Inverse:
.
Subtraction
- Definition:
.
Multiplication
- Commutative law:
.
- Associative law:
.
- Multiplicative identity:
.
- Multiplicative inverse:
, whenever
- Distributive law:
.
Division
- Definition:
, whenever
.
Let's look at an example to see how these rules are used in practice.
![]() |
= ![]() |
= ![]() |
|
= ![]() |
|
= ![]() |
Of course, the above is much longer than simply cancelling out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:
.
The correct simplification is
,
where the number cancels out in both the numerator and the denominator.
Interval notation
There are a few different ways that one can express with symbols a specific interval (all the numbers between two numbers). One way is with inequalities. If we wanted to denote the set of all numbers between, say, 2 and 4, we could write "all x satisfying 2<x<4." This excludes the endpoints 2 and 4 because we use < instead of . If we wanted to include the endpoints, we would write "all x satisfying
." This includes the endpoints.
Another way to write these intervals would be with interval notation. If we wished to convey "all x satisfying 2<x<4" we would write (2,4). This does not include the endpoints 2 and 4. If we wanted to include the endpoints we would write [2,4]. If we wanted to include 2 and not 4 we would write [2,4); if we wanted to exclude 2 and include 4, we would write (2,4].
Thus, we have the following table:
Endpoint conditions | Inequality notation | Interval notation |
---|---|---|
Including both 2 and 4 | all x satisfying ![]() |
![]() |
Not including 2 nor 4 | all x satisfying ![]() |
![]() |
Including 2 not 4 | all x satisfying ![]() |
![]() |
Including 4 not 2 | all x satisfying ![]() |
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In general, we have the following table:
Meaning | Interval Notation | Set Notation |
---|---|---|
All values greater than or equal to ![]() ![]() |
![]() |
![]() |
All values greater than ![]() ![]() |
![]() |
![]() |
All values greater than or equal to ![]() ![]() |
![]() |
![]() |
All values greater than ![]() ![]() |
![]() |
![]() |
All values greater than or equal to ![]() |
![]() |
![]() |
All values greater than ![]() |
![]() |
![]() |
All values less than or equal to ![]() |
![]() |
![]() |
All values less than ![]() |
![]() |
![]() |
All values. | ![]() |
![]() |
Note that and
must always have an exclusive parenthesis rather than an inclusive bracket. This is because
is not a number, and therefore cannot be in our set.
is really just a symbol that makes things easier to write, like the intervals above.
The interval (a,b) is called an open interval, and the interval [a,b] is called a closed interval.
Intervals are sets and we can use set notation to show relations between values and intervals. If we want to say that a certain value is contained in an interval, we can use the symbol to denote this. For example,
. Likewise, the symbol
denotes that a certain element is not in an interval. For example
.
Exponents and radicals
There are a few rules and properties involving exponents and radicals that you'd do well to remember. As a definition we have that if n is a positive integer then denotes n factors of a. That is,

If then we say that
.
If n is a negative integer then we say that
If we have an exponent that is a fraction then we say that
In addition to the previous definitions, the following rules apply:
Rule | Example |
---|---|
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Factoring and roots
Given the expression , one may ask "what are the values of x that make this expression 0?" If we factor we obtain

If x=-1 or -2, then one of the factors on the right becomes zero. Therefore, the whole must be zero. So, by factoring we have discovered the values of x that render the expression zero. These values are termed "roots." In general, given a quadratic polynomial that factors as

then we have that x = -c/a and x = -d/b are roots of the original polynomial.
A special case to be on the look out for is the difference of two squares, . In this case, we are always able to factor as

For example, consider . On initial inspection we would see that both
and
are squares (
and
). Applying the previous rule we have

The following is a general result of great utility.
The quadratic formula
Given any quadratic equation , all solutions of the equation are given by the quadratic formula:

Example: Find all the roots of
![]() Finding the roots is equivalent to solving the equation |
The quadratic formula can also help with factoring, as the next example demonstrates.
Example: Factor the polynomial
![]() We already know from the previous example that the polynomial has roots |
Note that if then the roots will not be real numbers.
Simplifying rational expressions
Consider the two polynomials

and

When we take the quotient of the two we obtain

The ratio of two polynomials is called a rational expression. Many times we would like to simplify such a beast. For example, say we are given We may simplify this in the following way:

This is nice because we have obtained something we understand quite well, , from something we didn't.
Formulas of multiplication of polynomials
Here are some formulas that can be quite useful for solving polynomial problems:





Polynomial Long Division
Suppose we would like to divide one polynomial by another. The procedure is similar to long division of numbers and is illustrated in the following example:
Example
Divide
![]() ![]() Similar to long division of numbers, we set up our problem as follows: First we have to answer the question, how many times does , and we multiply Now we perform the subtraction, bringing down any terms in the dividend that aren't matched in our subtrahend: Now we repeat, treating the bottom line as our new dividend: In this case we have no remainder. |
Application: Factoring Polynomials
We can use polynomial long division to factor a polynomial if we know one of the factors in advance. For example, suppose we have a polynomial and we know that
is a root of
. If we perform polynomial long division using P(x) as the dividend and
as the divisor, we will obtain a polynomial
such that
, where the degree of
is one less than the degree of
.
Exercise


Application: Breaking up a rational function
Similar to the way one can convert an improper fraction into an integer plus a proper fraction, one can convert a rational function whose numerator
has degree
and whose denominator
has degree
with
into a polynomial plus a rational function whose numerator has degree
and denominator has degree
with
.
Suppose that divided by
has quotient
and remainder
. That is
Dividing both sides by gives
will have degree less than
.
Example
Write
![]() so |
<h1> 1.2 Functions</h1>
What functions are and how are they described
Note: This is an attempt at a rewrite of "Classical understanding of functions". If others approve, consider deleting that section.
Whenever one quantity depends on one or more quantities, we have a function. You can think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product based on a specific set of instructions.
A function in everyday life
Think about dropping a ball from a bridge. At each moment in time, the ball is a height above the ground. The height of the ball is a function of time. It was the job of physicists to come up with a formula for this function. This type of function is called real-valued since the "finished product" is a number (or, more specifically, a real number). |
A function in everyday life (Preview of Multivariable Calculus)
Think about a wind storm. At different places, the wind can be blowing in different directions with different intensities. The direction and intensity of the wind can be thought of as a function of position. This is a function of two real variables (a location is described by two values - an |
We will be looking at real-valued functions until studying multivariable calculus. Think of a real-valued function as an input-output machine; you give the function an input, and it gives you an output which is a number (more specifically, a real number). For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input -1 and gives the output value 1.
There are many ways which people describe functions. In the examples above, a verbal descriptions is given (the height of the ball above the earth as a function of time). Here is a list of ways to describe functions. The top three listed approaches to describing functions are the most popular and you could skip the rest if you like.
- A function is given a name (such as
) and a formula for the function is also given. For example,
describes a function. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument.
- A function is described using an equation and two variables. One variable is for the input of the function and one is for the output of the function. The variable for the input is called the independent variable. The variable for the output is called the dependent variable. For example,
describes a function. The dependent variable appears by itself on the left hand side of equal sign.
- A verbal description of the function.
When a function is given a name (like in number 1 above), the name of the function is usually a single letter of the alphabet (such as or
). Some functions whose names are multiple letters (like the sine function
.
Plugging a value into a function
If we write
How would we know the value of the function and we would write
The value of Note that |
Classical understanding of functions
To provide the classical understanding of functions, think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product based on a specific set of instructions. The kinds of functions we consider here, for the most part, take in a real number, change it in a formulaic way, and give out a real number (possibly the same as the one it took in). Think of this as an input-output machine; you give the function an input, and it gives you an output. For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input and gives the output value 1.
A function is usually written as ,
, or something similar - although it doesn't have to be. A function is always defined as "of a variable" which tells us what to replace in the formula for the function.
For example, tells us:
- The function
is a function of
.
- To evaluate the function at a certain number, replace the
with that number.
- Replacing
with that number in the right side of the function will produce the function's output for that certain input.
- In English, the definition of
is interpreted, "Given a number,
will return two more than the triple of that number."
Thus, if we want to know the value (or output) of the function at 3:
We evaluate the function at
.
The value of
at 3 is 11.
See? It's easy!
Note that means the value of the dependent variable when
takes on the value of 3. So we see that the number 11 is the output of the function when we give the number 3 as the input. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument (or the dependent variable). A good way to think of it is the dependent variable
'depends' on the value of the independent variable
. This is read as "the value of
at three is eleven", or simply "
of three equals eleven".
Notation
Functions are used so much that there is a special notation for them. The notation is somewhat ambiguous, so familiarity with it is important in order to understand the intention of an equation or formula.
Though there are no strict rules for naming a function, it is standard practice to use the letters ,
, and
to denote functions, and the variable
to denote an independent variable.
is used for both dependent and independent variables.
When discussing or working with a function , it's important to know not only the function, but also its independent variable
. Thus, when referring to a function
, you usually do not write
, but instead
. The function is now referred to as "
of
". The name of the function is adjacent to the independent variable (in parentheses). This is useful for indicating the value of the function at a particular value of the independent variable. For instance, if
,
and if we want to use the value of for
equal to
, then we would substitute 2 for
on both sides of the definition above and write
This notation is more informative than leaving off the independent variable and writing simply '', but can be ambiguous since the parentheses can be misinterpreted as multiplication.
Modern understanding of functions
The formal definition of a function states that a function is actually a rule that associates elements of one set called the domain of the function, with the elements of another set called the range of the function. For each value we select from the domain of the function, there exists exactly one corresponding element in the range of the function. The definition of the function tells us which element in the range corresponds to the element we picked from the domain. Classically, the element picked from the domain is pictured as something that is fed into the function and the corresponding element in the range is pictured as the output. Since we "pick" the element in the domain whose corresponding element in the range we want to find, we have control over what element we pick and hence this element is also known as the "independent variable". The element mapped in the range is beyond our control and is "mapped to" by the function. This element is hence also known as the "dependent variable", for it depends on which independent variable we pick. Since the elementary idea of functions is better understood from the classical viewpoint, we shall use it hereafter. However, it is still important to remember the correct definition of functions at all times.
To make it simple, for the function , all of the possible
values constitute the domain, and all of the values
(
on the x-y plane) constitute the range.
Remarks
The following arise as a direct consequence of the definition of functions:
- By definition, for each "input" a function returns only one "output", corresponding to that input. While the same output may correspond to more than one input, one input cannot correspond to more than one output. This is expressed graphically as the vertical line test: a line drawn parallel to the axis of the dependent variable (normally vertical) will intersect the graph of a function only once. However, a line drawn parallel to the axis of the independent variable (normally horizontal) may intersect the graph of a function as many times as it likes. Equivalently, this has an algebraic (or formula-based) interpretation. We can always say if
, then
, but if we only know that
then we can't be sure that
.
- Each function has a set of values, the function's domain, which it can accept as input. Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}. This set must be implicitly/explicitly defined in the definition of the function. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element.
- Each function has a set of values, the function's range, which it can output. This may be the set of real numbers. It may be the set of positive integers or even the set {0,1}. This set, too, must be implicitly/explicitly defined in the definition of the function.
The vertical line test
The vertical line test, mentioned in the preceding paragraph, is a systematic test to find out if an equation involving and
can serve as a function (with
the independent variable and
the dependent variable). Simply graph the equation and draw a vertical line through each point of the
-axis. If any vertical line ever touches the graph at more than one point, then the equation is not a function; if the line always touches at most one point of the graph, then the equation is a function.
(There are a lot of useful curves, like circles, that aren't functions (see picture). Some people call these graphs with multiple intercepts, like our circle, "multi-valued functions"; they would refer to our "functions" as "single-valued functions".)
Important functions
Constant function | ![]() It disregards the input and always outputs the constant |
|
Linear function | ![]() Takes an input, multiplies by m and adds c. It is a polynomial of the first degree. Its graph is a line (slanted, except |
|
Identity function | ![]() Takes an input and outputs it unchanged. A polynomial of the first degree, f(x) = x1 = x. Special case of a linear function. |
|
Quadratic function | ![]() A polynomial of the second degree. Its graph is a parabola, unless |
|
Polynomial function | ![]() The number |
|
Signum function | ![]() Determines the sign of the argument |
Example functions
Some more simple examples of functions have been listed below.
![]()
|
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|
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|
It is possible to replace the independent variable with any mathematical expression, not just a number. For instance, if the independent variable is itself a function of another variable, then it could be replaced with that function. This is called composition, and is discussed later.
Manipulating functions
Addition, Subtraction, Multiplication and Division of functions
For two real-valued functions, we can add the functions, multiply the functions, raised to a power, etc.
Example: Adding, subtracting, multiplying and dividing functions which do not have a name
If we add the functions
|
If a math problem wants you to add two functions and
, there are two ways that the problem will likely be worded:
- If you are told that
, that
, that
and asked about
, then you are being asked to add two functions. Your answer would be
.
- If you are told that
, that
and you are asked about
, then you are being asked to add two functions. The addition of
and
is called
. Your answer would be
.
Similar statements can be made for subtraction, multiplication and division.
Example: Adding, subtracting, multiplying and dividing functions which do have a name
Let
|
Composition of functions
We begin with a fun (and not too complicated) application of composition of functions before we talk about what composition of functions is.
Example: Dropping a ball
If we drop a ball from a bridge which is 20 meters above the ground, then the height of our ball above the earth is a function of time. The physicists tell us that if we measure time in seconds and distance in meters, then the formula for height in terms of time is |
Composition of functions is another way to combine functions which is different from addition, subtraction, multiplication or division.
The value of a function depends upon the value of another variable
; however, that variable could be equal to another function
, so its value depends on the value of a third variable. If this is the case, then the first variable is a function
of the third variable; this function (
) is called the composition of the other two functions (
and
).
Example: Composing two functions
Let Let Then
Sometimes a math problem asks you compute Here,
|
Composition of functions is very common, mainly because functions themselves are common. For instance, squaring and sine are both functions:
,
Thus, the expression is a composition of functions:
-
= = .
(Note that this is not the same as .) Since the function sine equals
if
,
.
Since the function square equals if
,
.
Transformations
Transformations are a type of function manipulation that are very common. They consist of multiplying, dividing, adding or subtracting constants to either the input or the output. Multiplying by a constant is called dilation and adding a constant is called translation. Here are a few examples:
Dilation
Translation
Dilation
Translation
Translations and dilations can be either horizontal or vertical. Examples of both vertical and horizontal translations can be seen at right. The red graphs represent functions in their 'original' state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphs have been translated vertically.
Dilations are demonstrated in a similar fashion. The function
has had its input doubled. One way to think about this is that now any change in the input will be doubled. If I add one to , I add two to the input of
, so it will now change twice as quickly. Thus, this is a horizontal dilation by
because the distance to the
-axis has been halved. A vertical dilation, such as
is slightly more straightforward. In this case, you double the output of the function. The output represents the distance from the -axis, so in effect, you have made the graph of the function 'taller'. Here are a few basic examples where
is any positive constant:
Original graph | ![]() |
Rotation about origin | ![]() |
Horizontal translation by ![]() |
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Horizontal translation by ![]() |
![]() |
Horizontal dilation by a factor of ![]() |
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Vertical dilation by a factor of ![]() |
![]() |
Vertical translation by ![]() |
![]() |
Vertical translation by ![]() |
![]() |
Reflection about ![]() |
![]() |
Reflection about ![]() |
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Domain and Range
Domain
The domain of a function is the set of all points over which it is defined. More simply, it represents the set of x-values which the function can accept as input. For instance, if
then is only defined for values of
between
and
, because the square root function is not defined (in real numbers) for negative values. Thus, the domain, in interval notation, is
. In other words,
.
Range
The range of a function is the set of all values which it attains (i.e. the y-values). For instance, if:
,
then can only equal values in the interval from
to
. Thus, the range of
is
.
One-to-one Functions
A function is one-to-one (or less commonly injective) if, for every value of
, there is only one value of
that corresponds to that value of
. For instance, the function
is not one-to-one, because both
and
result in
. However, the function
is one-to-one, because, for every possible value of
, there is exactly one corresponding value of
. Other examples of one-to-one functions are
, where
. Note that if you have a one-to-one function and translate or dilate it, it remains one-to-one. (Of course you can't multiply
or
by a zero factor).
Horizontal Line Test
If you know what the graph of a function looks like, it is easy to determine whether or not the function is one-to-one. If every horizontal line intersects the graph in at most one point, then the function is one-to-one. This is known as the Horizontal Line Test.
Algebraic 1-1 Test
You can also show one-to-oneness algebraically by assuming that two inputs give the same output and then showing that the two inputs must have been equal. For example, Is a 1-1 function?
Therefore by the algebraic 1-1 test, the function is 1-1.
You can show that a function is not one-to-one by finding two distinct inputs that give the same output. For example, is not one-to-one because
but
.
Inverse functions
We call the inverse function of
if, for all
:
.
A function has an inverse function if and only if
is one-to-one. For example, the inverse of
is
. The function
has no inverse.
Notation
The inverse function of is denoted as
. Thus,
is defined as the function that follows this rule
:
To determine when given a function
, substitute
for
and substitute
for
. Then solve for
, provided that it is also a function.
Example: Given , find
.
Substitute for
and substitute
for
. Then solve for
:
To check your work, confirm that :
If isn't one-to-one, then, as we said before, it doesn't have an inverse. Then this method will fail.
Example: Given , find
.
Substitute for
and substitute
for
. Then solve for
:
Since there are two possibilities for , it's not a function. Thus
doesn't have an inverse. Of course, we could also have found this out from the graph by applying the Horizontal Line Test. It's useful, though, to have lots of ways to solve a problem, since in a specific case some of them might be very difficult while others might be easy. For example, we might only know an algebraic expression for
but not a graph.
<h1> 1.3 Graphing linear functions</h1>
It is sometimes difficult to understand the behavior of a function given only its definition; a visual representation or graph can be very helpful. A graph is a set of points in the Cartesian plane, where each point (,
) indicates that
. In other words, a graph uses the position of a point in one direction (the vertical-axis or y-axis) to indicate the value of
for a position of the point in the other direction (the horizontal-axis or x-axis).
Functions may be graphed by finding the value of for various
and plotting the points (
,
) in a Cartesian plane. For the functions that you will deal with, the parts of the function between the points can generally be approximated by drawing a line or curve between the points. Extending the function beyond the set of points is also possible, but becomes increasingly inaccurate.
Example
Plotting points like this is laborious. Fortunately, many functions' graphs fall into general patterns. For a simple case, consider functions of the form

The graph of is a single line, passing through the point
with slope 3. Thus, after plotting the point, a straightedge may be used to draw the graph. This type of function is called linear and there are a few different ways to present a function of this type.
Slope-intercept form
When we see a function presented as

we call this presentation the slope-intercept form. This is because, not surprisingly, this way of writing a linear function involves the slope, m, and the y-intercept, b.
Point-slope form
If someone walks up to you and gives you one point and a slope, you can draw one line and only one line that goes through that point and has that slope. Said differently, a point and a slope uniquely determine a line. So, if given a point and a slope m, we present the graph as

We call this presentation the point-slope form. The point-slope and slope-intercept form are essentially the same. In the point-slope form we can use any point the graph passes through. Where as, in the slope-intercept form, we use the y-intercept, that is the point (0,b).
Calculating slope
If given two points, and
, we may then compute the slope of the line that passes through these two points. Remember, the slope is determined as "rise over run." That is, the slope is the change in y-values divided by the change in x-values. In symbols,

So now the question is, "what's and
?" We have that
and
. Thus,

Two-point form
Two points also uniquely determine a line. Given points and
, we have the equation

This presentation is in the two-point form. It is essentially the same as the point-slope form except we substitute the expression for m.
<h1> 1.4 Precalculus Cumulative Exercises</h1>
Algebra
Convert to interval notation










State the following intervals using set notation
![[3,4] \,](../../../upload.wikimedia.org/math/7/5/4/754867828f4da2744549663cc47923aa.png)





Which one of the following is a true statement?
Hint: the true statement is often referred to as the triangle inequality. Give examples where the other two are false.



Evaluate the following expressions





![\sqrt[3]{\frac{27}{8}}](../../../upload.wikimedia.org/math/c/d/c/cdc339e9ed8f3192d2aa4919648a87fa.png)


![\frac{\sqrt{27}}{\sqrt[3]{9}}](../../../upload.wikimedia.org/math/9/d/5/9d5b26e7021395a068fba060e4a05634.png)
Simplify the following








Find the roots of the following polynomials








Factor the following expressions



Simplify the following




Functions
52. Let .




53. Let ,
.
- a. Give formulae for












55. Consider the following function

56. Consider the following function

57. Consider the following function
When you find the answer, you can add it here by clicking "edit".
When you find the answer, you can add it here by clicking "edit".

When you find the answer, you can add it here by clicking "edit".
58. Consider the following function
When you find the answer, you can add it here by clicking "edit".
When you find the answer, you can add it here by clicking "edit".

When you find the answer, you can add it here by clicking "edit".
Graphing
Limits
<h1> 2.1 An Introduction to Limits</h1>
Intuitive Look
A limit looks at what happens to a function when the input approaches a certain value. The general notation for a limit is as follows:
This is read as "The limit of of
as
approaches
". We'll take up later the question of how we can determine whether a limit exists for
at
and, if so, what it is. For now, we'll look at it from an intuitive standpoint.
Let's say that the function that we're interested in is , and that we're interested in its limit as
approaches
. Using the above notation, we can write the limit that we're interested in as follows:
One way to try to evaluate what this limit is would be to choose values near 2, compute for each, and see what happens as they get closer to 2. This is implemented as follows:
![]() |
1.7 | 1.8 | 1.9 | 1.95 | 1.99 | 1.999 |
---|---|---|---|---|---|---|
![]() |
2.89 | 3.24 | 3.61 | 3.8025 | 3.9601 | 3.996001 |
Here we chose numbers smaller than 2, and approached 2 from below. We can also choose numbers larger than 2, and approach 2 from above:
![]() |
2.3 | 2.2 | 2.1 | 2.05 | 2.01 | 2.001 |
---|---|---|---|---|---|---|
![]() |
5.29 | 4.84 | 4.41 | 4.2025 | 4.0401 | 4.004001 |
We can see from the tables that as grows closer and closer to 2,
seems to get closer and closer to 4, regardless of whether
approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of
as
approaches 2 is 4, or, written in limit notation,
We could have also just substituted 2 into and evaluated:
. However, this will not work with all limits.
Now let's look at another example. Suppose we're interested in the behavior of the function as
approaches 2. Here's the limit in limit notation:
Just as before, we can compute function values as approaches 2 from below and from above. Here's a table, approaching from below:
![]() |
1.7 | 1.8 | 1.9 | 1.95 | 1.99 | 1.999 |
---|---|---|---|---|---|---|
![]() |
-3.333 | -5 | -10 | -20 | -100 | -1000 |
And here from above:
![]() |
2.3 | 2.2 | 2.1 | 2.05 | 2.01 | 2.001 |
---|---|---|---|---|---|---|
![]() |
3.333 | 5 | 10 | 20 | 100 | 1000 |
In this case, the function doesn't seem to be approaching a single value as approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). This is known as an infinite limit. Note that we cannot just substitute 2 into
and evaluate as we could with the first example, since we would be dividing by 0.
Both of these examples may seem trivial, but consider the following function:
This function is the same as
Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input.
In algebra, we would simply say that we can cancel the term , and then we have the function
. This, however, would be a bit dishonest; the function that we have now is not really the same as the one we started with, because it is defined when
, and our original function was specifically not defined when
. In algebra we were willing to ignore this difficulty because we had no better way of dealing with this type of function. Now, however, in calculus, we can introduce a better, more correct way of looking at this type of function. What we want is to be able to say that, although the function doesn't exist when
, it works almost as though it does. It may not get there, but it gets really, really close. That is,
. The only question that we have is: what do we mean by "close"?
Informal Definition of a Limit
As the precise definition of a limit is a bit technical, it is easier to start with an informal definition; we'll explain the formal definition later.
We suppose that a function is defined for
near
(but we do not require that it be defined when
).
We call the limit of
as
approaches
if
becomes close to
when
is close (but not equal) to
, and if there is no other value
with the same property..
When this holds we write
or
Notice that the definition of a limit is not concerned with the value of when
(which may exist or may not). All we care about are the values of
when
is close to
, on either the left or the right (i.e. less or greater).
Limit Rules
Now that we have defined, informally, what a limit is, we will list some rules that are useful for working with and computing limits. You will be able to prove all these once we formally define the fundamental concept of the limit of a function.
First, the constant rule states that if (that is,
is constant for all
) then the limit as
approaches
must be equal to
. In other words
Constant Rule for Limits
- If b and c are constants then
.
- Example:
Second, the identity rule states that if (that is,
just gives back whatever number you put in) then the limit of
as
approaches
is equal to
. That is,
Identity Rule for Limits
- If c is a constant then
.
- Example:
The next few rules tell us how, given the values of some limits, to compute others.
Operational Identities for Limits
Suppose that and
and that
is constant. Then
Notice that in the last rule we need to require that is not equal to zero (otherwise we would be dividing by zero which is an undefined operation).
These rules are known as identities; they are the scalar product, sum, difference, product, and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by a constant, we say that you are performing scalar multiplication.)
Using these rules we can deduce another. Namely, using the rule for products many times we get that
for a positive integer
.
This is called the power rule.
Examples
- Example 1
Find the limit .
We need to simplify the problem, since we have no rules about this expression by itself. We know from the identity rule above that . By the power rule,
. Lastly, by the scalar multiplication rule, we get
.
- Example 2
Find the limit .
To do this informally, we split up the expression, once again, into its components. As above,.
Also and
. Adding these together gives
.
- Example 3
Find the limit .
From the previous example the limit of the numerator is . The limit of the denominator is
As the limit of the denominator is not equal to zero we can divide. This gives
.
- Example 4
Find the limit .
We apply the same process here as we did in the previous set of examples;
.
We can evaluate each of these;
and
Thus, the answer is
.
- Example 5
Find the limit .
In this example, evaluating the result directly will result in a division by zero. While you can determine the answer experimentally, a mathematical solution is possible as well.
First, the numerator is a polynomial that may be factored:
Now, you can divide both the numerator and denominator by (x-2):
- Example 6
Find the limit .
To evaluate this seemingly complex limit, we will need to recall some sine and cosine identities. We will also have to use two new facts. First, if is a trigonometric function (that is, one of sine, cosine, tangent, cotangent, secant or cosecant) and is defined at
, then
.
Second, . This may be determined experimentally, or by applying L'Hôpital's rule, described later in the book.
To evaluate the limit, recognize that can be multiplied by
to obtain
which, by our trig identities, is
. So, multiply the top and bottom by
. (This is allowed because it is identical to multiplying by one.) This is a standard trick for evaluating limits of fractions; multiply the numerator and the denominator by a carefully chosen expression which will make the expression simplify somehow. In this case, we should end up with:
.
Our next step should be to break this up into by the product rule. As mentioned above,
.
Next, .
Thus, by multiplying these two results, we obtain 0.
We will now present an amazingly useful result, even though we cannot prove it yet. We can find the limit at of any polynomial or rational function, as long as that rational function is defined at
(so we are not dividing by zero). That is,
must be in the domain of the function.
If is a polynomial or rational function that is defined at
then
We already learned this for trigonometric functions, so we see that it is easy to find limits of polynomial, rational or trigonometric functions wherever they are defined. In fact, this is true even for combinations of these functions; thus, for example, .
The Squeeze Theorem
The Squeeze Theorem is very important in calculus, where it is typically used to find the limit of a function by comparison with two other functions whose limits are known.
It is called the Squeeze Theorem because it refers to a function whose values are squeezed between the values of two other functions
and
, both of which have the same limit
. If the value of
is trapped between the values of the two functions
and
, the values of
must also approach
.
Expressed more precisely:
Suppose that






Example: Compute . Note that the sine of any real number is in the interval
. That is,
for all
, and
for all
. If
is positive, we can multiply these inequalities by
and get
. If
is negative, we can similarly multiply the inequalities by the positive number
and get
. Putting these together, we can see that, for all nonzero
,
. But it's easy to see that
. So, by the Squeeze Theorem,
.
Finding Limits
Now, we will discuss how, in practice, to find limits. First, if the function can be built out of rational, trigonometric, logarithmic and exponential functions, then if a number is in the domain of the function, then the limit at
is simply the value of the function at
.
If is not in the domain of the function, then in many cases (as with rational functions) the domain of the function includes all the points near
, but not
itself. An example would be if we wanted to find
, where the domain includes all numbers besides 0.
In that case, in order to find we want to find a function
similar to
, except with the hole at
filled in. The limits of
and
will be the same, as can be seen from the definition of a limit. By definition, the limit depends on
only at the points where
is close to
but not equal to it, so the limit at
does not depend on the value of the function at
. Therefore, if
,
also. And since the domain of our new function
includes
, we can now (assuming
is still built out of rational, trigonometric, logarithmic and exponential functions) just evaluate it at
as before. Thus we have
.
In our example, this is easy; canceling the 's gives
, which equals
at all points except 0. Thus, we have
. In general, when computing limits of rational functions, it's a good idea to look for common factors in the numerator and denominator.
Lastly, note that the limit might not exist at all. There are a number of ways in which this can occur:
- "Gap"
- There is a gap (not just a single point) where the function is not defined. As an example, in
does not exist when
. There is no way to "approach" the middle of the graph. Note that the function also has no limit at the endpoints of the two curves generated (at
and
). For the limit to exist, the point must be approachable from both the left and the right.
- Note also that there is no limit at a totally isolated point on a graph.
- "Jump"
- If the graph suddenly jumps to a different level, there is no limit at the point of the jump. For example, let
be the greatest integer
. Then, if
is an integer, when
approaches
from the right
, while when
approaches
from the left
. Thus
will not exist.
- Vertical asymptote
- In
- the graph gets arbitrarily high as it approaches 0, so there is no limit. (In this case we sometimes say the limit is infinite; see the next section.)
- Infinite oscillation
- These next two can be tricky to visualize. In this one, we mean that a graph continually rises above and falls below a horizontal line. In fact, it does this infinitely often as you approach a certain
-value. This often means that there is no limit, as the graph never approaches a particular value. However, if the height (and depth) of each oscillation diminishes as the graph approaches the
-value, so that the oscillations get arbitrarily smaller, then there might actually be a limit.
- The use of oscillation naturally calls to mind the trigonometric functions. An example of a trigonometric function that does not have a limit as
approaches 0 is
- As
gets closer to 0 the function keeps oscillating between
and 1. In fact,
oscillates an infinite number of times on the interval between 0 and any positive value of
. The sine function is equal to zero whenever
, where
is a positive integer. Between every two integers
,
goes back and forth between 0 and
or 0 and 1. Hence,
for every
. In between consecutive pairs of these values,
and
,
goes back and forth from 0, to either
or 1 and back to 0. We may also observe that there are an infinite number of such pairs, and they are all between 0 and
. There are a finite number of such pairs between any positive value of
and
, so there must be infinitely many between any positive value of
and 0. From our reasoning we may conclude that, as
approaches 0 from the right, the function
does not approach any specific value. Thus,
does not exist.
Using Limit Notation to Describe Asymptotes
Now consider the function
What is the limit as approaches zero? The value of
does not exist; it is not defined.
Notice, also, that we can make as large as we like, by choosing a small
, as long as
. For example, to make
equal to
, we choose
to be
. Thus,
does not exist.
However, we do know something about what happens to when
gets close to 0 without reaching it. We want to say we can make
arbitrarily large (as large as we like) by taking
to be sufficiently close to zero, but not equal to zero. We express this symbolically as follows:
Note that the limit does not exist at ; for a limit, being
is a special kind of not existing. In general, we make the following definition.

We say the limit of as
approaches
is infinity if
becomes very big (as big as we like) when
is close (but not equal) to
.
In this case we write
or
.
Similarly, we say the limit of as
approaches
is negative infinity if
becomes very negative when
is close (but not equal) to
.
In this case we write
or
.
An example of the second half of the definition would be that .
Key Application of Limits
To see the power of the concept of the limit, let's consider a moving car. Suppose we have a car whose position is linear with respect to time (that is, a graph plotting the position with respect to time will show a straight line). We want to find the velocity. This is easy to do from algebra; we just take the slope, and that's our velocity.
But unfortunately, things in the real world don't always travel in nice straight lines. Cars speed up, slow down, and generally behave in ways that make it difficult to calculate their velocities.
Now what we really want to do is to find the velocity at a given moment (the instantaneous velocity). The trouble is that in order to find the velocity we need two points, while at any given time, we only have one point. We can, of course, always find the average speed of the car, given two points in time, but we want to find the speed of the car at one precise moment.
This is the basic trick of differential calculus, the first of the two main subjects of this book. We take the average speed at two moments in time, and then make those two moments in time closer and closer together. We then see what the limit of the slope is as these two moments in time are closer and closer, and say that this limit is the slope at a single instant.
We will study this process in much greater depth later in the book. First, however, we will need to study limits more carefully.
External Links
<h1> 2.2 Finite Limits</h1>
Informal Finite Limits
Now, we will try to more carefully restate the ideas of the last chapter. We said then that the equation meant that, when
gets close to 2,
gets close to 4. What exactly does this mean? How close is "close"? The first way we can approach the problem is to say that, at
,
, which is pretty close to 4.
Sometimes however, the function might do something completely different. For instance, suppose , so
. Next, if you take a value even closer to 2,
, in this case you actually move further from 4. The reason for this is that substitution gives us 4.23 as x approaches 2.
The solution is to find out what happens arbitrarily close to the point. In particular, we want to say that, no matter how close we want the function to get to 4, if we make close enough to 2 then it will get there. In this case, we will write
and say "The limit of , as
approaches 2, equals 4" or "As
approaches 2,
approaches 4." In general:
We call the limit of
as
approaches
if
becomes arbitrarily close to
whenever
is sufficiently close (and not equal) to
.
When this holds we write
or
One-Sided Limits
Sometimes, it is necessary to consider what happens when we approach an value from one particular direction. To account for this, we have one-sided limits. In a left-handed limit,
approaches
from the left-hand side. Likewise, in a right-handed limit,
approaches
from the right-hand side.
For example, if we consider , there is a problem because there is no way for
to approach 2 from the left hand side (the function is undefined here). But, if
approaches 2 only from the right-hand side, we want to say that
approaches 0.
We call the limit of
as
approaches
from the right if
becomes arbitrarily close to
whenever
is sufficiently close to and greater than
.
When this holds we write
Similarly, we call the limit of
as
approaches
from the left if
becomes arbitrarily close to
whenever
is sufficiently close to and less than
.
When this holds we write
In our example, the left-handed limit does not exist.
The right-handed limit, however, .
It is a fact that exists if and only if
and
exist and are equal to each other. In this case,
will be equal to the same number.
In our example, one limit does not even exist. Thus does not exist either.
<h1> 2.3 Infinite Limits</h1>
Informal Infinite Limits
Another kind of limit involves looking at what happens to as
gets very big. For example, consider the function
. As
gets very big,
gets very small. In fact,
gets closer and closer to zero the bigger
gets. Without limits it is very difficult to talk about this fact, because
can keep getting bigger and bigger and
never actually gets to zero; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something - without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does
have to be to be sure that
is really going towards 0?
In this case, we want to say that, however close we want to get to 0, for
big enough
is guaranteed to get that close. So we have yet another definition.
We call the limit of
as
approaches infinity if
becomes arbitrarily close to
whenever
is sufficiently large.
When this holds we write
or
Similarly, we call the limit of
as
approaches negative infinity if
becomes arbitrarily close to
whenever
is sufficiently negative.
When this holds we write
or
So, in this case, we write:
and say "The limit, as approaches infinity, equals
," or "as
approaches infinity, the function approaches
".
We can also write:
because making very negative also forces
to be close to
.
Notice, however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.
Limits at Infinity of Rational Functions
One special case that comes up frequently is when we want to find the limit at (or
) of a rational function. A rational function is just one made by dividing two polynomials by each other. For example,
is a rational function. Also, any polynomial is a rational function, since
is just a (very simple) polynomial, so we can write the function
as
, the quotient of two polynomials.
Consider the numerator of a rational function as we allow the variable to grow very large (in either the positive or negative sense). The term with the highest exponent on the variable will dominate the numerator, and the other terms become more and more insignificant compared to the dominating term. The same applies to the denominator. In the limit, the other terms become negligible, and we only need to examine the dominating term in the numerator and denominator.
There is a simple rule for determining a limit of a rational function as the variable approaches infinity. Look for the term with the highest exponent on the variable in the numerator. Look for the same in the denominator. This rule is based on that information.
- If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit (at both
and
) is the ratio of the coefficients of the highest terms.
- If the numerator has the highest term, then the fraction is called "top-heavy". If, when you divide the numerator by the denominator the resulting exponent on the variable is even, then the limit (at both
and
) is
. If it is odd, then the limit at
is
, and the limit at
is
.
- If the denominator has the highest term, then the fraction is called "bottom-heavy" and the limit (at both
and
) is zero.
Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same as . Also, a straight power of
, like
, has coefficient 1, since it is the same as
.
Examples
- Example 1
Find .
The function is the quotient of two polynomials,
and
. By our rule we look for the term with highest exponent in the numerator; it's
. The term with highest exponent in the denominator is also
. So, the limit is the ratio of their coefficients. Since
, both coefficients are 1, so
.
- Example 2
Find .
We look at the terms with the highest exponents; for the numerator, it is , while for the denominator it is
. Since the exponent on the numerator is higher, we know the limit at
will be
. So,
.
Infinity is not a number
Most people seem to struggle with this fact when first introduced to calculus, and in particular limits.
But is different.
is not a number.
Mathematics is based on formal rules that govern the subject. When a list of formal rules applies to a type of object (e.g., "a number") those rules must always apply — no exceptions!
What makes different is this: "there is no number greater than infinity". You can write down the formula in a lot of different ways, but here's one way:
. If you add one to infinity, you still have infinity; you don't have a bigger number. If you believe that, then infinity is not a number.
Since does not follow the rules laid down for numbers, it cannot be a number. Every time you use the symbol
in a formula where you would normally use a number, you have to interpret the formula differently. Let's look at how
does not follow the rules that every actual number does:
Addition Breaks
Every number has a negative, and addition is associative. For we could write
and note that
. This is a good thing, since it means we can prove if you take one away from infinity, you still have infinity:
. But it also means we can prove 1 = 0, which is not so good.
Therefore, .
Reinterpret Formulas that Use 
We started off with a formula that does "mean" something, even though it used and
is not a number.
What does this mean, compared to what it means when we have a regular number instead of an infinity symbol:
This formula says that I can make sure the values of don't differ very much from
, so long as I can control how much x varies away from 2. I don't have to make
exactly equal to
, but I also can't control x too tightly. I have to give you a range to vary x within. It's just going to be very, very small (probably) if you want to make
very very close to
. And by the way, it doesn't matter at all what happens when
.
If we could use the same paragraph as a template for my original formula, we'll see some problems. Let's substitute 0 for 2, and for
.
This formula says that I can make sure the values of don't differ very much from
, so long as I can control how much x varies away from 0. I don't have to make
exactly equal to
, but I also can't control x too tightly. I have to give you a range to vary x within. It's just going to be very, very small (probably) if you want to see that
gets very, very close to
. And by the way, it doesn't matter at all what happens when
.
It's close to making sense, but it isn't quite there. It doesn't make sense to say that some real number is really "close" to . For example, when
and
does it really makes sense to say 1000 is closer to
than 1 is? Solve the following equations for δ:
No real number is very close to ; that's what makes
so special! So we have to rephrase the paragraph:
This formula says that I can make sure the values of get as big as any number you pick, so long as I can control how much x varies away from 0. I don't have to make
bigger than every number, but I also can't control x too tightly. I have to give you a range to vary x within. It's just going to be very, very small (probably) if you want to see that
gets very, very large. And by the way, it doesn't matter at all what happens when
.
You can see that the essential nature of the formula hasn't changed, but the exact details require some human interpretation. While rigorous definitions and clear distinctions are essential to the study of mathematics, sometimes a bit of casual rewording is okay. You just have to make sure you understand what a formula really means so you can draw conclusions correctly.
Exercises
Write out an explanatory paragraph for the following limits that include . Remember that you will have to change any comparison of magnitude between a real number and
to a different phrase. In the second case, you will have to work out for yourself what the formula means.


<h1> 2.4 Continuity</h1>
Defining Continuity
We are now ready to define the concept of a function being continuous. The idea is that we want to say that a function is continuous if you can draw its graph without taking your pencil off the page. But sometimes this will be true for some parts of a graph but not for others. Therefore, we want to start by defining what it means for a function to be continuous at one point. The definition is simple, now that we have the concept of limits:
If is defined on an open interval containing
, then
is said to be continuous at
if and only if

Note that for to be continuous at
, the definition in effect requires three conditions:
- that
is defined at
, so
exists,
- the limit as
approaches
exists, and
- the limit and
are equal.
If any of these do not hold then is not continuous at
.
The idea of the definition is that the point of the graph corresponding to will be close to the points of the graph corresponding to nearby
-values. Now we can define what it means for a function to be continuous in general, not just at one point.
A function is said to be continuous on


We often use the phrase "the function is continuous" to mean that the function is continuous at every real number. This would be the same as saying the function was continuous on (−∞, ∞), but it is a bit more convenient to simply say "continuous".
Note that, by what we already know, the limit of a rational, exponential, trigonometric or logarithmic function at a point is just its value at that point, so long as it's defined there. So, all such functions are continuous wherever they're defined. (Of course, they can't be continuous where they're not defined!)
Discontinuities
A discontinuity is a point where a function is not continuous. There are lots of possible ways this could happen, of course. Here we'll just discuss two simple ways.
Removable discontinuities
The function is not continuous at
. It is discontinuous at that point because the fraction then becomes
, which is undefined. Therefore the function fails the first of our three conditions for continuity at the point 3; 3 is just not in its domain.
However, we say that this discontinuity is removable. This is because, if we modify the function at that point, we can eliminate the discontinuity and make the function continuous. To see how to make the function continuous, we have to simplify
, getting
. We can define a new function
where
. Note that the function
is not the same as the original function
, because
is defined at
, while
is not. Thus,
is continuous at
, since
. However, whenever
,
; all we did to
to get
was to make it defined at
.
In fact, this kind of simplification is often possible with a discontinuity in a rational function. We can divide the numerator and the denominator by a common factor (in our example ) to get a function which is the same except where that common factor was 0 (in our example at
). This new function will be identical to the old except for being defined at new points where previously we had division by 0.
However, this is not possible in every case. For example, the function has a common factor of
in both the numerator and denominator, but when you simplify you are left with
, which is still not defined at
. In this case the domain of
and
are the same, and they are equal everywhere they are defined, so they are in fact the same function. The reason that
differed from
in the first example was because we could take it to have a larger domain and not simply that the formulas defining
and
were different.
Jump discontinuities
Not all discontinuities can be removed from a function. Consider this function:
Since does not exist, there is no way to redefine
at one point so that it will be continuous at 0. These sorts of discontinuities are called nonremovable discontinuities.
Note, however, that both one-sided limits exist; and
. The problem is that they are not equal, so the graph "jumps" from one side of 0 to the other. In such a case, we say the function has a jump discontinuity. (Note that a jump discontinuity is a kind of nonremovable discontinuity.)
One-Sided Continuity
Just as a function can have a one-sided limit, a function can be continuous from a particular side. For a function to be continuous at a point from a given side, we need the following three conditions:
- the function is defined at the point,
- the function has a limit from that side at that point and
- the one-sided limit equals the value of the function at the point.
A function will be continuous at a point if and only if it is continuous from both sides at that point. Now we can define what it means for a function to be continuous on a closed interval.
A function is said to be continuous on if and only if
- it is continuous on
,
- it is continuous from the right at
and
- it is continuous from the left at
.
Notice that, if a function is continuous, then it is continuous on every closed interval contained in its domain.
Intermediate Value Theorem
A useful theorem regarding continuous functions is the following:
If a function

![[a,b]](../../../upload.wikimedia.org/math/2/c/3/2c3d331bc98b44e71cb2aae9edadca7e.png)







Application: bisection method
The bisection method is the simplest and most reliable algorithm to find zeros of a continuous function.
Suppose we want to solve the equation . Given two points
and
such that
and
have opposite signs, the intermediate value theorem tells us that
must have at least one root between
and
as long as
is continuous on the interval
. If we know
is continuous in general (say, because it's made out of rational, trigonometric, exponential and logarithmic functions), then this will work so long as
is defined at all points between
and
. So, let's divide the interval
in two by computing
. There are now three possibilities:
,
and
have opposite signs, or
and
have opposite signs.
In the first case, we're done. In the second and third cases, we can repeat the process on the sub-interval where the sign change occurs. In this way we hone in to a small sub-interval containing the zero. The midpoint of that small sub-interval is usually taken as a good approximation to the zero.
Note that, unlike the methods you may have learned in algebra, this works for any continuous function that you (or your calculator) know how to compute.
<h1> 2.5 Formal Definition of the Limit</h1>
In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (after all, it took mathematicians 150 years to arrive at it); it is also the most important and most useful.
The intuitive definition of a limit is inadequate to prove anything rigorously about it. The problem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of this term is that the closer gets to the specified value, the closer the function must get to the limit, so that however close we want the function to the limit, we can accomplish this by making
sufficiently close to our value. We can express this requirement technically as follows:
Let be a function defined on an open interval
that contains
, except possibly at
. Let
be a number. Then we say that
if, for every , there exists a
such that for all
with
we have
.
To further explain, earlier we said that "however close we want the function to the limit, we can find a corresponding close to our value." Using our new notation of epsilon (
) and delta (
), we mean that if we want to make
within
of
, the limit, then we know that making
within
of
puts it there.
Again, since this is tricky, let's resume our example from before: , at
. To start, let's say we want
to be within .01 of the limit. We know by now that the limit should be 4, so we say: for
, there is some
so that as long as
, then
To show this, we can pick any that is bigger than 0, so long as it works. For example, you might pick
, because you are absolutely sure that if
is within .00000000000001 of 2, then
will be within .01 of 4. This
works for
. But we can't just pick a specific value for
, like .01, because we said in our definition "for every
." This means that we need to be able to show an infinite number of
s, one for each
. We can't list an infinite number of
s!
Of course, we know of a very good way to do this; we simply create a function, so that for every , it can give us a
. In this case, one definition of
that works is
(see example 5 in choosing delta for an explanation of how this delta was chosen.)
So, in general, how do you show that tends to
as
tends to
? Well imagine somebody gave you a small number
(e.g., say
). Then you have to find a
and show that whenever
we have
. Now if that person gave you a smaller
(say
) then you would have to find another
, but this time with 0.03 replaced by 0.002. If you can do this for any choice of
then you have shown that
tends to
as
tends to
. Of course, the way you would do this in general would be to create a function giving you a
for every
, just as in the example above.
Formal Definition of the Limit at Infinity
We call the limit of
as
approaches
if for every number
there exists a
such that whenever
we have
When this holds we write
or
as
Similarly, we call the limit of
as
approaches
if for every number
, there exists a number
such that whenever
we have
When this holds we write
or
as
Notice the difference in these two definitions. For the limit of as
approaches
we are interested in those
such that
. For the limit of
as
approaches
we are interested in those
such that
.
Examples
Here are some examples of the formal definition.
Example 1
We know from earlier in the chapter that
.
What is when
for this limit?
We start with the desired conclusion and substitute the given values for and
:
.
Then we solve the inequality for :
This is the same as saying
.
(We want the thing in the middle of the inequality to be because that's where we're taking the limit.) We normally choose the smaller of
and
for
, so
, but any smaller number will also work.
Example 2
What is the limit of as
approaches 4?
There are two steps to answering such a question; first we must determine the answer — this is where intuition and guessing is useful, as well as the informal definition of a limit — and then we must prove that the answer is right.
In this case, 11 is the limit because we know is a continuous function whose domain is all real numbers. Thus, we can find the limit by just substituting 4 in for
, so the answer is
.
We're not done, though, because we never proved any of the limit laws rigorously; we just stated them. In fact, we couldn't have proved them, because we didn't have the formal definition of the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove that no matter what value of is given to us, we can find a value of
such that
whenever
For this particular problem, letting works (see choosing delta for help in determining the value of
to use in other problems). Now, we have to prove
given that
.
Since , we know
which is what we wished to prove.
Example 3
What is the limit of as
approaches 4?
As before, we use what we learned earlier in this chapter to guess that the limit is . Also as before, we pull out of thin air that
.
Note that, since is always positive, so is
, as required. Now, we have to prove
given that
.
We know that
(because of the triangle inequality), so
Example 4
Show that the limit of as
approaches 0 does not exist.
We will proceed by contradiction. Suppose the limit exists; call it . For simplicity, we'll assume that
; the case for
is similar. Choose
. Then if the limit were
there would be some
such that
for every
with
. But, for every
, there exists some (possibly very large)
such that
, but
, a contradiction.
Example 5
What is the limit of as
approaches 0?
By the Squeeze Theorem, we know the answer should be 0. To prove this, we let . Then for all
, if
, then
as required.
Example 6
Suppose that and
. What is
?
Of course, we know the answer should be , but now we can prove this rigorously. Given some
, we know there's a
such that, for any
with
,
(since the definition of limit says "for any
", so it must be true for
as well). Similarly, there's a
such that, for any
with
,
. We can set
to be the lesser of
and
. Then, for any
with
,
, as required.
If you like, you can prove the other limit rules too using the new definition. Mathematicians have already done this, which is how we know the rules work. Therefore, when computing a limit from now on, we can go back to just using the rules and still be confident that our limit is correct according to the rigorous definition.
Formal Definition of a Limit Being Infinity
Let be a function defined on an open interval
that contains
, except possibly at
. Then we say that
if, for every , there exists a
such that for all
with
we have
.
When this holds we write
or
as
.
Similarly, we say that
if, for every , there exists a
such that for all
with
we have
.
When this holds we write
or
as
.
<h1> 2.6 Proofs of Some Basic Limit Rules</h1>
Now that we have the formal definition of a limit, we can set about proving some of the properties we stated earlier in this chapter about limits.
Constant Rule for Limits
- If b and c are constants then
.
Proof of the Constant Rule for Limits:
To prove that , we need to find a
such that for every
,
whenever
.
and
, so
is satisfied independent of any value of
; that is, we can choose any
we like and the
condition holds.
Identity Rule for Limits
- If c is a constant then
.
Proof of the Identity Rule for Limits:
To prove that , we need to find a
such that for every
,
whenever
. Choosing
satisfies this condition.
Scalar Product Rule for Limits




Proof of the Scalar Product Rule for Limits:
Since we are given that , there must be some function, call it
, such that for every
,
whenever
. Now we need to find a
such that for all
,
whenever
.
First let's suppose that .
, so
. In this case, letting
satisfies the limit condition.
Now suppose that . Since
has a limit at
, we know from the definition of a limit that
is defined in an open interval D that contains
(except maybe at
itself). In particular, we know that
doesn't blow up to infinity within D (except maybe at
, but that won't affect the limit), so that
in D. Since
is the constant function
in D, the limit
by the Constant Rule for Limits.
Finally, suppose that .
, so
. In this case, letting
satisfies the limit condition.
Sum Rule for Limits
Suppose that and
. Then
![\lim_{x\to c} [f(x) + g(x)] = \lim_{x\to c} f(x) + \lim_{x\to c} g(x) = L + M](../../../upload.wikimedia.org/math/f/c/9/fc91ee0a02d49e26d14348dde1d463dd.png)
Proof of the Sum Rule for Limits:
Since we are given that and
, there must be functions, call them
and
, such that for all
,
whenever
, and .
whenever
.
Adding the two inequalities gives . By the triangle inequality we have
, so we have
whenever
and
. Let
be the smaller of
and
. Then this
satisfies the definition of a limit for
having limit
.
Difference Rule for Limits
Suppose that and
. Then
![\lim_{x\to c} [f(x) - g(x)] = \lim_{x\to c} f(x) - \lim_{x\to c} g(x) = L - M](../../../upload.wikimedia.org/math/1/e/f/1ef531ea9f08039695124590bc484671.png)
Proof of the Difference Rule for Limits: Define . By the Scalar Product Rule for Limits,
. Then by the Sum Rule for Limits,
.
Product Rule for Limits
Suppose that and
. Then
![\lim_{x\to c} [f(x) g(x)] = \lim_{x\to c} f(x) \lim_{x\to c} g(x) = L M](../../../upload.wikimedia.org/math/6/7/6/676e5368f616024e2ed8163257486ea8.png)
Proof of the Product Rule for Limits:[1]
Let be any positive number. The assumptions imply the existence of the positive numbers
such that
when
when
when
According to the condition (3) we see that
when
Supposing then that and using (1) and (2) we obtain
Quotient Rule for Limits
Suppose that and
and
. Then

Proof of the Quotient Rule for Limits:
If we can show that , then we can define a function,
as
and appeal to the Product Rule for Limits to prove the theorem. So we just need to prove that
.
Let be any positive number. The assumptions imply the existence of the positive numbers
such that
when
when
According to the condition (2) we see that
when
which implies that
when
Supposing then that and using (1) and (3) we obtain
Suppose that






Proof of the Squeeze Theorem:
From the assumptions, we know that there exists a such that
and
when
.
These inequalities are equivalent to and
when
.
Using what we know about the relative ordering of , and
, we have
when
.
or
when
.
So
when
.
Notes
- ↑ This proof is adapted from one found at planetmath.org/encyclopedia/ProofOfLimitRuleOfProduct.html due to Planet Math user pahio and made available under the terms of the Creative Commons By/Share-Alike License.
<h1> 2.7 Limits Cumulative Exercises</h1>
Basic Limit Exercises


One-Sided Limits
Evaluate the following limits or state that the limit does not exist.




Two-Sided Limits
Evaluate the following limits or state that the limit does not exist.




















Limits to Infinity
Evaluate the following limits or state that the limit does not exist.










Limits of Piecewise Functions
Evaluate the following limits or state that the limit does not exist.
37. Consider the function



38. Consider the function






39. Consider the function




External Links
Differentiation
Basics of Differentiation
<h1> 3.1 Differentiation Defined</h1>
What is Differentiation?
Differentiation is an operation that allows us to find a function that outputs the rate of change of one variable with respect to another variable.
Informally, we may suppose that we're tracking the position of a car on a two-lane road with no passing lanes. Assuming the car never pulls off the road, we can abstractly study the car's position by assigning it a variable, . Since the car's position changes as the time changes, we say that
is dependent on time, or
. This tells where the car is at each specific time. Differentiation gives us a function
which represents the car's speed, that is the rate of change of its position with respect to time.
Equivalently, differentiation gives us the slope at any point of the graph of a non-linear function. For a linear function, of form ,
is the slope. For non-linear functions, such as
, the slope can depend on
; differentiation gives us a function which represents this slope.
The Definition of Slope
Historically, the primary motivation for the study of differentiation was the tangent line problem: for a given curve, find the slope of the straight line that is tangent to the curve at a given point. The word tangent comes from the Latin word tangens, which means touching. Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope. But what exactly do we mean by "slope" for a curve?
The solution is obvious in some cases: for example, a line is its own tangent; the slope at any point is
. For the parabola
, the slope at the point
is
; the tangent line is horizontal.
But how can you find the slope of, say, at
? This is in general a nontrivial question, but first we will deal carefully with the slope of lines.
Of a line
The slope of a line, also called the gradient of the line, is a measure of its inclination. A line that is horizontal has slope 0, a line from the bottom left to the top right has a positive slope and a line from the top left to the bottom right has a negative slope.
The slope can be defined in two (equivalent) ways. The first way is to express it as how much the line climbs for a given motion horizontally. We denote a change in a quantity using the symbol (pronounced "delta"). Thus, a change in
is written as
. We can therefore write this definition of slope as:
An example may make this definition clearer. If we have two points on a line, and
, the change in
from
to
is given by:
Likewise, the change in from
to
is given by:
This leads to the very important result below.
The slope of the line between the points and
is
-
.
Alternatively, we can define slope trigonometrically, using the tangent function:
where is the angle from the rightward-pointing horizontal to the line, measured counter-clockwise. If you recall that the tangent of an angle is the ratio of the y-coordinate to the x-coordinate on the unit circle, you should be able to spot the equivalence here.
Of a graph of a function
The graphs of most functions we are interested in are not straight lines (although they can be), but rather curves. We cannot define the slope of a curve in the same way as we can for a line. In order for us to understand how to find the slope of a curve at a point, we will first have to cover the idea of tangency. Intuitively, a tangent is a line which just touches a curve at a point, such that the angle between them at that point is zero. Consider the following four curves and lines:
(i) | (ii) |
![]() |
![]() |
(iii) | (iv) |
![]() |
![]() |
- The line
crosses, but is not tangent to
at
.
- The line
crosses, and is tangent to
at
.
- The line
crosses
at two points, but is tangent to
only at
.
- There are many lines that cross
at
, but none are tangent. In fact, this curve has no tangent at
.
A secant is a line drawn through two points on a curve. We can construct a definition of a tangent as the limit of a secant of the curve taken as the separation between the points tends to zero. Consider the diagram below.
As the distance tends to zero, the secant line becomes the tangent at the point
. The two points we draw our line through are:
and
As a secant line is simply a line and we know two points on it, we can find its slope, , using the formula from before:
(We will refer to the slope as because it may, and generally will, depend on
.) Substituting in the points on the line,
This simplifies to
This expression is called the difference quotient. Note that can be positive or negative — it is perfectly valid to take a secant through any two points on the curve — but cannot be
.
The definition of the tangent line we gave was not rigorous, since we've only defined limits of numbers — or, more precisely, of functions that output numbers — not of lines. But we can define the slope of the tangent line at a point rigorously, by taking the limit of the slopes of the secant lines from the last paragraph. Having done so, we can then define the tangent line as well. Note that we cannot simply set to zero as this would imply division of zero by zero which would yield an undefined result. Instead we must find the limit of the above expression as
tends to zero:
The slope of the graph of at the point
is
If this limit does not exist, then we say the slope is undefined.
If the slope is defined, say , then the tangent line to the graph of
at the point
is the line with equation
This last equation is just the point-slope form for the line through with slope
.
Exercises


The Rate of Change of a Function at a Point
Consider the formula for average velocity in the direction,
, where
is the change in
over the time interval
. This formula gives the average velocity over a period of time, but suppose we want to define the instantaneous velocity. To this end we look at the change in position as the change in time approaches 0. Mathematically this is written as:
, which we abbreviate by the symbol
. (The idea of this notation is that the letter
denotes change.) Compare the symbol
with
. The idea is that both indicate a difference between two numbers, but
denotes a finite difference while
denotes an infinitesimal difference. Please note that the symbols
and
have no rigorous meaning on their own, since
, and we can't divide by 0.
(Note that the letter is often used to denote distance, which would yield
. The letter
is often avoided in denoting distance due to the potential confusion resulting from the expression
.)
The Definition of the Derivative
You may have noticed that the two operations we've discussed — computing the slope of the tangent to the graph of a function and computing the instantaneous rate of change of the function — involved exactly the same limit. That is, the slope of the tangent to the graph of is
. Of course,
can, and generally will, depend on
, so we should really think of it as a function of
. We call this process (of computing
) differentiation. Differentiation results in another function whose value for any value
is the slope of the original function at
. This function is known as the derivative of the original function.
Since lots of different sorts of people use derivatives, there are lots of different mathematical notations for them. Here are some:
(read "f prime of x") for the derivative of
,
,
,
for the derivative of
as a function of
or
, which is more useful in some cases.
Most of the time the brackets are not needed, but are useful for clarity if we are dealing with something like , where we want to differentiate the product of two functions,
and
.
The first notation has the advantage that it makes clear that the derivative is a function. That is, if we want to talk about the derivative of at
, we can just write
.
In any event, here is the formal definition:
Let






Examples
Example 1
The derivative of is
,
no matter what is. This is consistent with the definition of the derivative as the slope of a function.
Example 2
What is the slope of the graph of at
? We can do it "the hard (and imprecise) way", without using differentiation, as follows, using a calculator and using small differences below and above the given point:
When ,
.
When ,
.
Then the difference between the two values of is
.
Then the difference between the two values of is
.
Thus, the slope at the point of the graph at which
.
But, to solve the problem precisely, we compute
-
= = = = = = .
We were lucky this time; the approximation we got above turned out to be exactly right. But this won't always be so, and, anyway, this way we didn't need a calculator.
In general, the derivative of is
-
= = = = = = = .
Example 3
If (the absolute value function) then
, which can also be stated as
Finding this derivative is a bit complicated, so we won't prove it at this point.
Here, is not smooth (though it is continuous) at
and so the limits
and
(the limits as 0 is approached from the right and left respectively) are not equal. From the definition,
, which does not exist. Thus,
is undefined, and so
has a discontinuity at 0. This sort of point of non-differentiability is called a cusp. Functions may also not be differentiable because they go to infinity at a point, or oscillate infinitely frequently.
Understanding the derivative notation
The derivative notation is special and unique in mathematics. The most common notation for derivatives you'll run into when first starting out with differentiating is the Leibniz notation, expressed as . You may think of this as "rate of change in
with respect to
". You may also think of it as "infinitesimal value of
divided by infinitesimal value of
". Either way is a good way of thinking, although you should remember that the precise definition is the one we gave above. Often, in an equation, you will see just
, which literally means "derivative with respect to x". This means we should take the derivative of whatever is written to the right; that is,
means
where
.
As you advance through your studies, you will see that we sometimes pretend that and
are separate entities that can be multiplied and divided, by writing things like
. Eventually you will see derivatives such as
, which just means that the input variable of our function is called
and our output variable is called
; sometimes, we will write
, to mean the derivative with respect to
of whatever is written on the right. In general, the variables could be anything, say
.
All of the following are equivalent for expressing the derivative of
Exercises














Differentiation Rules
The process of differentiation is tedious for complicated functions. Therefore, rules for differentiating general functions have been developed, and can be proved with a little effort. Once sufficient rules have been proved, it will be fairly easy to differentiate a wide variety of functions. Some of the simplest rules involve the derivative of linear functions.
Derivative of a constant function
For any fixed real number ,
![\frac{d}{dx}\left[c\right]=0.](../../../upload.wikimedia.org/math/e/7/8/e78275207f52d546f36ed2fe57f21068.png)
Intuition
The graph of the function is a horizontal line, which has a constant slope of zero. Therefore, it should be expected that the derivative of this function is zero, regardless of the values of
and
.
Proof
The definition of a derivative is
Let for all
. (That is,
is a constant function.) Then
. Therefore
.
Let . To prove that
, we need to find a positive
such that, for any given positive
,
whenever
. But
, so
for any choice of
.
Examples
Note that, in the second example, is just a constant.
Derivative of a linear function
For any fixed real numbers and
,
![\frac{d}{dx}\left[mx+c\right]=m](../../../upload.wikimedia.org/math/5/e/6/5e6c935ca3b1af98f89e72822c568415.png)
The special case shows the advantage of the
notation—rules are intuitive by basic algebra, though this does not constitute a proof, and can lead to misconceptions to what exactly
and
actually are.
Intuition
The graph of is a line with constant slope
.
Proof
If , then
. So,
-
= = = = =
Constant multiple and addition rules
Since we already know the rules for some very basic functions, we would like to be able to take the derivative of more complex functions by breaking them up into simpler functions. Two tools that let us do this are the constant multiple rule and the addition rule.
The Constant Rule
For any fixed real number ,
![\frac{d}{dx}\left[cf(x)\right] = c \frac{d}{dx}\left[f(x)\right]](../../../upload.wikimedia.org/math/4/3/a/43a914dad1efe3423b921e4dd6924280.png)
The reason, of course, is that one can factor out of the numerator, and then of the entire limit, in the definition. The details are left as an exercise.
Example
We already know that
.
Suppose we want to find the derivative of
-
= = =
Another simple rule for breaking up functions is the addition rule.
The Addition and Subtraction Rules
![\frac{d}{dx}\left[f(x)\pm g(x)\right]= \frac{d}{dx}\left[f(x)\right]\pm\frac{d}{dx}\left[g(x)\right]](../../../upload.wikimedia.org/math/6/e/0/6e0b689997ec5729e484327f7db5a807.png)
Proof
From the definition:
By definition then, this last term is
Example
What is the derivative of ?
-
= = = =
The fact that both of these rules work is extremely significant mathematically because it means that differentiation is linear. You can take an equation, break it up into terms, figure out the derivative individually and build the answer back up, and nothing odd will happen.
We now need only one more piece of information before we can take the derivatives of any polynomial.
The Power Rule
![\frac{d}{dx}\left[x^n\right]=nx^{n-1}](../../../upload.wikimedia.org/math/d/3/e/d3e5c4f744a44c47de9fc6c92e7137e0.png)
For example, in the case of the derivative is
as was established earlier. A special case of this rule is that
.
Since polynomials are sums of monomials, using this rule and the addition rule lets you differentiate any polynomial. A relatively simple proof for this can be derived from the binomial expansion theorem.
This rule also applies to fractional and negative powers. Therefore
-
= = =
Derivatives of polynomials
With these rules in hand, you can now find the derivative of any polynomial you come across. Rather than write the general formula, let's go step by step through the process.
The first thing we can do is to use the addition rule to split the equation up into terms:
We can immediately use the linear and constant rules to get rid of some terms:
Now you may use the constant multiplier rule to move the constants outside the derivatives:
Then use the power rule to work with the individual monomials:
And then do some algebra to get the final answer:
These are not the only differentiation rules. There are other, more advanced, differentiation rules, which will be described in a later chapter.
Exercises
- Find the derivatives of the following equations:



<h1> 3.2 Product and Quotient Rules</h1>
Product Rule
When we wish to differentiate a more complicated expression such as
our only way (up to this point) to differentiate the expression is to expand it and get a polynomial, and then differentiate that polynomial. This method becomes very complicated and is particularly error prone when doing calculations by hand. A beginner might guess that the derivative of a product is the product of the derivatives, similar to the sum and difference rules, but this is not true. To take the derivative of a product, we use the product rule.
|
It may also be stated as
or in the Leibniz notation as
.
The derivative of the product of three functions is:
.
Since the product of two or more functions occurs in many mathematical models of physical phenomena, the product rule has broad application in physics, chemistry, and engineering.
Examples
- Suppose one wants to differentiate ƒ(x) = x2 sin(x). By using the product rule, one gets the derivative ƒ '(x) = 2x sin(x) + x2cos(x) (since the derivative of x2 is 2x and the derivative of sin(x) is cos(x)).
- One special case of the product rule is the constant multiple rule, which states: if c is a real number and ƒ(x) is a differentiable function, then cƒ(x) is also differentiable, and its derivative is (c × ƒ)'(x) = c × ƒ '(x). This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear.
Physics Example I: rocket acceleration
Consider the vertical acceleration of a model rocket relative to its initial position at a fixed point on the ground. Newton's second law says that the force is equal to the time rate change of momentum. If F is the net force (sum of forces), p is the momentum, and t is the time,
Since the momentum is equal to the product of mass and velocity, this yields
where m is the mass and v is the velocity. Application of the product rule gives
Since the acceleration, a, is defined as the time rate change of velocity, a = dv/dt,
Solving for the acceleration,
Since the rocket is losing mass, dm/dt is negative, and the changing mass term results in increased acceleration.[1][2]
Physics Example II: electromagnetic induction
Faraday's law of electromagnetic induction states that the induced electromotive force is the negative time rate of change of magnetic flux through a conducting loop.
where is the electromotive force (emf) in volts and ΦB is the magnetic flux in webers. For a loop of area, A, in a magnetic field, B, the magnetic flux is given by
where θ is the angle between the normal to the current loop and the magnetic field direction.
Taking the negative derivative of the flux with respect to time yields the electromotive force gives
In many cases of practical interest only one variable (A, B, or θ) is changing, so two of the three above terms are often zero.
Proof of the Product Rule
Proving this rule is relatively straightforward, first let us state the equation for the derivative:
We will then apply one of the oldest tricks in the book—adding a term that cancels itself out to the middle:
Notice that those terms sum to zero, and so all we have done is add 0 to the equation. Now we can split the equation up into forms that we already know how to solve:
Looking at this, we see that we can separate the common terms out of the numerators to get:
Which, when we take the limit, becomes:
, or the mnemonic "one D-two plus two D-one"
This can be extended to 3 functions:
For any number of functions, the derivative of their product is the sum, for each function, of its derivative times each other function.
Back to our original example of a product, , we find the derivative by the product rule is
Note, its derivative would not be
which is what you would get if you assumed the derivative of a product is the product of the derivatives.
To apply the product rule we multiply the first function by the derivative of the second and add to that the derivative of first function multiply by the second function. Sometimes it helps to remember the memorize the phrase "First times the derivative of the second plus the second times the derivative of the first."
Application: proof of the Power Rule
The product rule can be used to give a proof of the power rule for whole numbers. The proof proceeds by mathematical induction. We begin with the base case . If
then from the definition is easy to see that
Next we suppose that for fixed value of , we know that for
,
. Consider the derivative of
,
We have shown that the statement is true for
and that if this statement holds for
, then it also holds for
. Thus by the principle of mathematical induction, the statement must hold for
.
Quotient Rule
There is a similar rule for quotients. To prove it, we go to the definition of the derivative:
This leads us to the so-called "quotient rule":
|
Some people remember this rule with the mnemonic "low D-high minus high D-low, over the square of what's below!"
Examples
The derivative of is:
Remember: the derivative of a product/quotient is not the product/quotient of the derivatives. (That is, differentiation does not distribute over multiplication or division.) However one can distribute before taking the derivative. That is
References
<h1> 3.3 Derivatives of Trigonometric Functions</h1>
Sine, cosine, tangent, cosecant, secant, cotangent. These are functions that crop up continuously in mathematics and engineering and have a lot of practical applications. They also appear in more advanced mathematics, particularly when dealing with things such as line integrals with complex numbers and alternate representations of space like spherical and cylindrical coordinate systems.
We use the definition of the derivative, i.e.,
,
to work these first two out.
Let us find the derivative of sin x, using the above definition.
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|
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Definition of derivative |
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trigonometric identity |
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factoring |
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separation of terms |
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application of limit |
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solution |
Now for the case of cos x.
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|
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Definition of derivative |
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trigonometric identity |
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factoring |
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separation of terms |
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application of limit |
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solution |
Therefore we have established
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To find the derivative of the tangent, we just remember that:
which is a quotient. Applying the quotient rule, we get:
Then, remembering that , we simplify:
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For secants, we again apply the quotient rule.
Leaving us with:
Simplifying, we get:
|
Using the same procedure on cosecants:
We get:
|
Using the same procedure for the cotangent that we used for the tangent, we get:
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<h1> 3.4 Chain Rule</h1>
The chain rule is a method to compute the derivative of the functional composition of two or more functions.
If a function, f, depends on a variable, u, which in turn depends on another variable, x, that is f = y(u(x)) , then the rate of change of f with respect to x can be computed as the rate of change of y with respect to u multiplied by the rate of change of u with respect to x.
If a function f is composed to two differentiable functions y(x) and u(x), so that f(x) = y(u(x)), then f(x) is differentiable and, |
The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another.[1] For example, if f is a function of g which is in turn a function of h, which is in turn a function of x, that is
,
the derivative of f with respect to x is given by
and so on.
A useful mnemonic is to think of the differentials as individual entities that you can cancel algebraically, such as
However, keep in mind that this trick comes about through a clever choice of notation rather than through actual algebraic cancellation.
The chain rule has broad applications in physics, chemistry, and engineering, as well as being used to study related rates in many disciplines. The chain rule can also be generalized to multiple variables in cases where the nested functions depend on more than one variable.
Examples
Example I
Suppose that a mountain climber ascends at a rate of 0.5 kilometer per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometer. To calculate the decrease in air temperature per hour that the climber experiences, one multiplies 6 °C per kilometer by 0.5 kilometer per hour, to obtain 3 °C per hour. This calculation is a typical chain rule application.
Example II
Consider the function f(x) = (x2 + 1)3. It follows from the chain rule that
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Function to differentiate |
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Define u(x) as inside function |
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Express f(x) in terms of u(x) |
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Express chain rule applicable here |
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Substitute in f(u) and u(x) |
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Compute derivatives with power rule |
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Substitute u(x) back in terms of x |
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Simplify. |
Example III
In order to differentiate the trigonometric function
one can write:
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Function to differentiate |
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Define u(x) as inside function |
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Express f(x) in terms of u(x) |
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Express chain rule applicable here |
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Substitute in f(u) and u(x) |
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Evaluate derivatives |
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Substitute u in terms of x. |
Example IV: absolute value
The chain rule can be used to differentiate , the absolute value function:
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Function to differentiate |
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Equivalent function |
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Define u(x) as inside function |
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Express f(x) in terms of u(x) |
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Express chain rule applicable here |
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Substitute in f(u) and u(x) |
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Compute derivatives with power rule |
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Substitute u(x) back in terms of x |
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Simplify |
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Express ![]() |
Example V: three nested functions
The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another. For example, if , sequential application of the chain rule yields the derivative as follows (we make use of the fact that
, which will be proved in a later section):
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Original (outermost) function |
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Define h(x) as innermost function |
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g(h) = sin(h) as middle function |
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Express chain rule applicable here |
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Differentiate f(g)[2] |
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Differentiate g(h) |
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Differentiate h(x) |
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Substitute into chain rule. |
Chain Rule in Physics
Because one physical quantity often depends on another, which, in turn depends on others, the chain rule has broad applications in physics. This section presents examples of the chain rule in kinematics and simple harmonic motion. The chain rule is also useful in electromagnetic induction.
Physics Example I: relative kinematics of two vehicles
For example, one can consider the kinematics problem where one vehicle is heading west toward an intersection at 80 miles per hour while another is heading north away from the intersection at 60 miles per hour. One can ask whether the vehicles are getting closer or further apart and at what rate at the moment when the northbound vehicle is 3 miles north of the intersection and the westbound vehicle is 4 miles east of the intersection.
Big idea: use chain rule to compute rate of change of distance between two vehicles.
Plan:
- Choose coordinate system
- Identify variables
- Draw picture
- Big idea: use chain rule to compute rate of change of distance between two vehicles
- Express c in terms of x and y via Pythagorean theorem
- Express dc/dt using chain rule in terms of dx/dt and dy/dt
- Substitute in x, y, dx/dt, dy/dt
- Simplify.
Choose coordinate system: Let the y-axis point north and the x-axis point east.
Identify variables: Define y(t) to be the distance of the vehicle heading north from the origin and x(t) to be the distance of the vehicle heading west from the origin.
Express c in terms of x and y via Pythagorean theorem:
Express dc/dt using chain rule in terms of dx/dt and dy/dt:
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Apply derivative operator to entire function |
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Sum of squares is inside function |
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Distribute differentiation operator |
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Apply chain rule to x(t) and y(t)} |
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Simplify. |
Substitute in x = 4 mi, y = 3 mi, dx/dt = −80 mi/hr, dy/dt = 60 mi/hr and simplify
Consequently, the two vehicles are getting closer together at a rate of 28 mi/hr.
Physics Example II: harmonic oscillator
If the displacement of a simple harmonic oscillator from equilibrium is given by x, and it is released from its maximum displacement A at time t = 0, then the position at later times is given by
where ω = 2 π/T is the angular frequency and T is the period of oscillation. The velocity, v, being the first time derivative of the position can be computed with the chain rule:
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Definition of velocity in one dimension |
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Substitute x(t) |
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Bring constant A outside of derivative |
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Differentiate outside function (cosine) |
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Bring negative sign in front |
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Evaluate remaining derivative |
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Simplify. |
The acceleration is then the second time derivative of position, or simply dv/dt.
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Definition of acceleration in one dimension |
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Substitute v(t) |
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Bring constant term outside of derivative |
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Differentiate outside function (sine) |
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Evaluate remaining derivative |
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Simplify. |
From Newton's second law, F = ma, where F is the net force and m is the object's mass.
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Newton's second law |
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Substitute a(t) |
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Simplify |
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Substitute original x(t). |
Thus it can be seen that these results are consistent with the observation that the force on a simple harmonic oscillator is a negative constant times the displacement.
Chain Rule in Chemistry
The chain rule has many applications in Chemistry because many equations in Chemistry describe how one physical quantity depends on another, which in turn depends on another. For example, the ideal gas law describes the relationship between pressure, volume, temperature, and number of moles, all of which can also depend on time.
Chemistry Example I: Ideal Gas Law
Suppose a sample of n moles of an ideal gas is held in an isothermal (constant temperature, T) chamber with initial volume V0. The ideal gas is compressed by a piston so that its volume changes at a constant rate so that V(t) = V0 - kt, where t is the time. The chain rule can be employed to find the time rate of change of the pressure.[3] The ideal gas law can be solved for the pressure, P to give:
where P(t) and V(t) have been written as explicit functions of time and the other symbols are constant. Differentiating both sides yields
where the constant terms, n, R, and T, have been moved to the left of the derivative operator. Applying the chain rule gives
where the power rule has been used to differentiate 1/V, Since V(t) = V0 - kt, dV/dt = -k. Substituting in for V and dV/dt yields dP/dt.
Chemistry Example II: Kinetic Theory of Gases

A second application of the chain rule in Chemistry is finding the rate of change of the average molecular speed, v, in an ideal gas as the absolute temperature T, increases at a constant rate so that T = T0 + at, where T0 is the initial temperature and t is the time.[3] The kinetic theory of gases relates the root mean square of the molecular speed to the temperature, so that if v(t) and T(t) are functions of time,
where R is the ideal gas constant, and M is the molecular weight.
Differentiating both sides with respect to time yields:
Using the chain rule to express the right side in terms of the with respect to temperature, T, and time, t, respectively gives
Evaluating the derivative with respect to temperature, T, yields
Evaluating the remaining derivative with respect to T, taking the reciprocal of the negative power, and substituting T = T0 + at, produces
Evaluating the derivative with respect to t yields
which simplifies to
Exercises







References
External links
<h1> 3.5 Higher Order Derivatives</h1>
The second derivative, or second order derivative, is the derivative of the derivative of a function. The derivative of the function may be denoted by
, and its double (or "second") derivative is denoted by
. This is read as "f double prime of x," or "The second derivative of f(x)." Because the derivative of function
is defined as a function representing the slope of function
, the double derivative is the function representing the slope of the first derivative function.
Furthermore, the third derivative is the derivative of the derivative of the derivative of a function, which can be represented by . This is read as "f triple prime of x", or "The third derivative of f(x)". This can continue as long as the resulting derivative is itself differentiable, with the fourth derivative, the fifth derivative, and so on. Any derivative beyond the first derivative can be referred to as a higher order derivative.
Notation
Let be a function in terms of x. The following are notations for higher order derivatives.
2nd Derivative | 3rd Derivative | 4th Derivative | nth Derivative | Notes |
---|---|---|---|---|
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Probably the most common notation. |
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Leibniz notation. |
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Another form of Leibniz notation. |
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Euler's notation. |
Warning: You should not write to indicate the nth derivative, as this is easily confused with the quantity
all raised to the nth power.
The Leibniz notation, which is useful because of its precision, follows from
.
Newton's dot notation extends to the second derivative, , but typically no further in the applications where this notation is common.
Examples
Example 1:
Find the third derivative ofwith respect to x.
Repeatedly apply the Power Rule to find the derivatives.
Example 2:
Find the third derivative ofwith respect to x.
Applications:
For applications of the second derivative in finding a curve's concavity and points of inflection, see "Extrema and Points of Inflection" and "Extreme Value Theorem". For applications of higher order derivatives in physics, see the "Kinematics" section.
<h1> Failed to match page to section number. Check your argument; if correct, consider updating Template:Calculus/map page. Implicit Differentiation</h1>
Generally, you will encounter functions expressed in explicit form, that is, in the form . To find the derivative of y with respect to x, you take the derivative with respect to x of both sides of the equation to get
But suppose you have a relation of the form . In this case, it may be inconvenient or even impossible to solve for y as a function of x. A good example is the relation
. In this case you can utilize implicit differentiation to find the derivative. To do so, one takes the derivative of both sides of the equation with respect to x and solves for
. That is, form
and solve for dy/dx. You need to employ the chain rule whenever you take the derivative of a variable with respect to a different variable. For example,
Implicit Differentiation and the Chain Rule
To understand how implicit differentiation works and use it effectively it is important to recognize that the key idea is simply the chain rule. First let's recall the chain rule. Suppose we are given two differentiable functions f(x) and g(x) and that we are interested in computing the derivative of the function f(g(x)), the the chain rule states that:
That is, we take the derivative of f as normal and then plug in g, finally multiply the result by the derivative of g.
Now suppose we want to differentiate a term like y2 with respect to x where we are thinking of y as a function of x, so for the remainder of this calculation let's write it as y(x) instead of just y. The term y2 is just the composition of f(x) = x2 and y(x). That is, f(y(x)) = y2(x). Recalling that f′(x) = 2x then the chain rule states that:
Of course it is customary to think of y as being a function of x without always writing y(x), so this calculation usually is just written as
Don't be confused by the fact that we don't yet know what y′ is, it is some function and often if we are differentiating two quantities that are equal it becomes possible to explicitly solve for y′ (as we will see in the examples below.) This makes it a very powerful technique for taking derivatives.
Explicit Differentiation
For example, suppose we are interested in the derivative of y with respect to x where x and y are related by the equation
This equation represents a circle of radius 1 centered on the origin. Note that y is not a function of x since it fails the vertical line test ( when
, for example).
To find y', first we can separate variables to get
Taking the square root of both sides we get two separate functions for y:
We can rewrite this as a fractional power:
Using the chain rule we get,
And simplifying by substituting y back into this equation gives
Implicit Differentiation
Using the same equation
First, differentiate with respect to x on both sides of the equation:
To differentiate the second term on the left hand side of the equation (call it f(y(x))=y2), use the chain rule:
So the equation becomes
Separate the variables:
Divide both sides by , and simplify to get the same result as above:
Uses
Implicit differentiation is useful when differentiating an equation that cannot be explicitly differentiated because it is impossible to isolate variables.
For example, consider the equation,
Differentiate both sides of the equation (remember to use the product rule on the term xy):
Isolate terms with y':
Factor out a y' and divide both sides by the other term:
Example
can be solved as:
then differentiated:
However, using implicit differentiation it can also be differentiated like this:
use the product rule:
solve for :
Note that, if we substitute into
, we end up with
again.
Application: inverse trigonometric functions
Arcsine, arccosine, arctangent. These are the functions that allow you to determine the angle given the sine, cosine, or tangent of that angle.
First, let us start with the arcsine such that:
To find dy/dx we first need to break this down into a form we can work with:
Then we can take the derivative of that:
...and solve for dy / dx:
At this point we need to go back to the unit triangle. Since y is the angle and the opposite side is sin(y) (which is equal to x), the adjacent side is cos(y) (which is equal to the square root of 1 minus x2, based on the Pythagorean theorem), and the hypotenuse is 1. Since we have determined the value of cos(y) based on the unit triangle, we can substitute it back in to the above equation and get:
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We can use an identical procedure for the arccosine and arctangent:
|
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<h1> 3.7 Derivatives of Exponential and Logarithm Functions</h1>
Exponential Function
First, we determine the derivative of using the definition of the derivative:
Then we apply some basic algebra with powers (specifically that ab + c = ab ac):
Since ex does not depend on h, it is constant as h goes to 0. Thus, we can use the limit rules to move it to the outside, leaving us with:
Now, the limit can be calculated by techniques we will learn later, for example Calculus/L'Hôpital's rule, and we will see that
so that we have proved the following rule:
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Now that we have derived a specific case, let us extend things to the general case. Assuming that a is a positive real constant, we wish to calculate:
One of the oldest tricks in mathematics is to break a problem down into a form that we already know we can handle. Since we have already determined the derivative of ex, we will attempt to rewrite ax in that form.
Using that eln(c) = c and that ln(ab) = b · ln(a), we find that:
Thus, we simply apply the chain rule:
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Logarithm Function
Closely related to the exponentiation is the logarithm. Just as with exponents, we will derive the equation for a specific case first (the natural log, where the base is e), and then work to generalize it for any logarithm.
First let us create a variable y such that:
It should be noted that what we want to find is the derivative of y or .
Next we will put both sides to the power of e in an attempt to remove the logarithm from the right hand side:
Now, applying the chain rule and the property of exponents we derived earlier, we take the derivative of both sides:
This leaves us with the derivative:
Substituting back our original equation of x = ey, we find that:
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If we wanted, we could go through that same process again for a generalized base, but it is easier just to use properties of logs and realize that:
Since 1 / ln(b) is a constant, we can just take it outside of the derivative:
Which leaves us with the generalized form of:
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Logarithmic Differentiation
We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such a products with many terms, quotients of composed functions, or functions with variable or function exponents. We do this by taking the natural logarithm of both sides, re-arranging terms using the logarithm laws below, and then differentiating both sides implicitly, before multiplying through by y.
|
See the examples below.
- Example 1
Suppose we wished to differentiate We take the natural logarithm of both sides Differentiating implicitly, recalling the chain rule Multiplying by y, the original function |
- Example 2
Let us differentiate a function Taking the natural logarithm of left and right We then differentiate both sides, recalling the product and chain rules Multiplying by the original function y |
- Example 3
Take a function Then We then differentiate And finally multiply by y |
<h1> 3.8 Some Important Theorems</h1>
This section covers three theorems of fundamental importance to the topic of differential calculus: The Extreme Value Theorem, Rolle's Theorem, and the Mean Value Theorem. It also discusses the relationship between differentiability and continuity.
Extreme Value Theorem
Classification of Extrema
We start out with some definitions.
A global maximum (also called an absolute maximum) of a function






A global minimum (also called an absolute minimum) of a function






Maxima and minima are collectively known as extrema.
The Extreme Value Theorem
If




The Extreme Value Theorem is a fundamental result of real analysis whose proof is beyond the scope of this text. However, the truth of the theorem allows us to talk about the maxima and minima of continuous functions on closed intervals without concerning ourselves with whether or not they exist. When dealing with functions that do not satisfy the premises of the theorem, we will need to worry about such things. For example, the unbounded function has no extrema whatsoever. If
is restricted to the semi-closed interval
[
), then
has a minimum value of
at
, but it has no maximum value since, for any given value
in
, one can always find a larger value of
for
in
, for example by forming
, where
is the average of
with
. The function
has a discontinuity at
.
fails to have any extrema in any closed interval around
since the function is unbounded below as one approaches
from the left, and it is unbounded above as one approaches
from the right. (In fact, the function is undefined for x=0. However, the example is unaffected if g(0) is assigned any arbitrary value.)
The Extreme Value Theorem is an existence theorem. It tells us that global extrema exist if certain conditions are met, but it doesn't tell us how to find them. We will discuss how to determine the extrema of continuous functions in the section titled Extrema and Points of Inflection.
Rolle's Theorem
If a function,

![[a,b] \](../../../upload.wikimedia.org/math/c/f/c/cfcbcc2c2a41716ab844e25069e87453.png)




Rolle's Theorem is important in proving the Mean Value Theorem. Intuitively it says that if you have a function that is continuous everywhere in an interval bounded by points where the function has the same value, and if the function is differentiable everywhere in the interval (except maybe at the endpoints themselves), then the function must have zero slope in at least one place in the interior of the interval.
Proof of Rolle's Theorem
If is constant on
, then
for every
in
, so the theorem is true. So for the remainder of the discussion we assume
is not constant on
.
Since satisfies the conditions of the Extreme Value Theorem,
must attain its maximum and minimum values on
. Since
is not constant on
, the endpoints cannot be both maxima and minima. Thus, at least one extremum exists in
. We can suppose without loss of generality that this extremum is a maximum because, if it were a minimum, we could consider the function
instead. Let
with
in
be a maximum. It remains to be shown that
.
By the definition of derivative, . By substituting
, this is equivalent to
. Note that
for all
in
since
is the maximum on
.
since it has non-positive numerator and negative denominator.
since it has non-positive numerator and positive denominator.
The limits from the left and right must be equal since the function is differentiable at , so
.
Exercise

Mean Value Theorem
If is continuous on the closed interval
and differentiable on the open interval
, there exists a number,
, in the open interval
such that
.
The Mean Value Theorem is an important theorem of differential calculus. It basically says that for a differentiable function defined on an interval, there is some point on the interval whose instantaneous slope is equal to the average slope of the interval. Note that Rolle's Theorem is the special case of the Mean Value Theorem when .
In order to prove the Mean Value Theorem, we will prove a more general statement, of which the Mean Value Theorem is a special case. The statement is Cauchy's Mean Value Theorem, also known as the Extended Mean Value Theorem.
Cauchy's Mean Value Theorem
If ,
are continuous on the closed interval
and differentiable on the open interval
, then there exists a number,
, in the open interval
such that
If and
, then this is equivalent to
.
To prove Cauchy's Mean Value Theorem, consider the function . Since both
and
are continuous on
and differentiable on
, so is
.
.Since
(see the exercises), Rolle's Theorem tells us that there exists some number
in
such that
. This implies that
, which is what was to be shown.
Exercises







![[0,\pi]](../../../upload.wikimedia.org/math/e/1/8/e1868564b62b4e2f1c063321df289469.png)
Differentiability Implies Continuity
If exists then
is continuous at
. To see this, note that
. But
This imples that or
, which shows that
is continuous at
.
The converse, however, is not true. Take , for example.
is continuous at 0 since
and
and
, but it is not differentiable at 0 since
but
.
<h1> 3.9 Basics of Differentiation Cumulative Exercises</h1>
Find the Derivative by Definition
Find the derivative of the following functions using the limit definition of the derivative.









Prove the Constant Rule

![\frac{d}{dx}\left[cf(x)\right] = c \frac{d}{dx}\left[f(x)\right]](../../../upload.wikimedia.org/math/4/3/a/43a914dad1efe3423b921e4dd6924280.png)
Find the Derivative by Rules
Find the derivative of the following functions:
Power Rule

![f(x) = 3\sqrt[3]{x}\,](../../../upload.wikimedia.org/math/c/b/b/cbb20f52c448cac11f99b9527d212b6e.png)




![f(x) = \frac{3}{x^4} - \sqrt[4]{x} + x \,](../../../upload.wikimedia.org/math/7/a/5/7a5e8965529063e1dcb09e2b4ebe0782.png)

![f(x) = \frac{1}{\sqrt[3]{x}} + \sqrt{x} \,](../../../upload.wikimedia.org/math/a/1/9/a19701b4473e0092c7007aa739181d46.png)
Product Rule









Quotient Rule














Chain Rule











Exponentials




Logarithms





Trigonometric functions


More Differentiation
![\frac{d}{dx}[(x^{3}+5)^{10}]](../../../upload.wikimedia.org/math/e/a/d/ead360cd62636e2512acbe4d656b2655.png)
![\frac{d}{dx}[x^{3}+3x]](../../../upload.wikimedia.org/math/d/b/c/dbce3ac485fac127f932d7a23e69537f.png)
![\frac{d}{dx}[(x+4)(x+2)(x-3)]](../../../upload.wikimedia.org/math/b/6/9/b69873a920a7a9f943285e11742f9add.png)
![\frac{d}{dx}[\frac{x+1}{3x^{2}}]](../../../upload.wikimedia.org/math/f/c/c/fcc667fbf7a0d6a3f6df15a53b8419bd.png)
![\frac{d}{dx}[3x^{3}]](../../../upload.wikimedia.org/math/6/1/9/619bbc93a4fb50b1c46c8fd61b88c25d.png)
![\frac{d}{dx}[x^{4}\sin x]](../../../upload.wikimedia.org/math/6/b/8/6b808ac8534dbed2d1eb509f63ee7340.png)
![\frac{d}{dx}[2^{x}]](../../../upload.wikimedia.org/math/1/d/c/1dc8558b501331d29cd6e45f43e7f17a.png)
![\frac{d}{dx}[e^{x^{2}}]](../../../upload.wikimedia.org/math/b/8/c/b8c0a852488e802d4d9094c91b00cfe4.png)
![\frac{d}{dx}[e^{2^{x}}]](../../../upload.wikimedia.org/math/1/a/5/1a52029c7d3bbc2eb45db4d453f1f0de.png)
Implicit Differentiation
Use implicit differentiation to find y'


Logarithmic Differentiation
Use logarithmic differentiation to find :
![y = x(\sqrt[4]{1-x^3}\,)](../../../upload.wikimedia.org/math/c/a/6/ca6f41c9f6005c02d1bed079be037495.png)




Equation of Tangent Line
For each function, , (a) determine for what values of
the tangent line to
is horizontal and (b) find an equation of the tangent line to
at the given point.








Higher Order Derivatives

External Links
Applications of Derivatives
<h1> Failed to match page to section number. Check your argument; if correct, consider updating Template:Calculus/map page. L'Hôpital's Rule</h1>
L'Hopital's Rule
Occasionally, one comes across a limit which results in or
, which are called indeterminate limits. However, it is still possible to solve these in many cases due to L'Hôpital's rule. This rule also is vital in explaining how a number of other limits can be derived.
If




All of the following expressions are indeterminate forms.
These expressions are called indeterminate because you cannot determine their exact value in the indeterminate form. Depending on the situation, each indeterminate form could evaluate to a variety of values.
Theorem
If
is indeterminate of type
or
,
then
In other words, if the limit of the function is indeterminate, the limit equals the derivative of the top over the derivative of the bottom. If that is indeterminate, L'Hôpital's rule can be used again until the limit isn't
or
.
Note:
can approach a finite value c,
or
.
Proof of the 0/0 case
Suppose that for real functions and
,
and
and that
exists. Thus
and
exist in an interval
around
, but maybe not at
itself. This implies that both
and
are differentiable (and thus continuous) everywhere in
except perhaps at
. Thus, for any
in
, in any interval
or
,
and
are continuous and differentiable, with the possible exception of
. Define
and
.
Note that ,
and that
and
are continuous in any interval
or
and differentiable in any interval
or
when
is in
. Cauchy's Mean Value Theorem tells us that
for some
in
(if
) or
(if
). Since
, we have
for
and
in
. Note that
is the same limit as
since both
and
are being squeezed to
. So taking the limit as
of the last equation gives
which is equivalent to
.
Examples
Example 1
Find
Since plugging in 0 for x results in , use L'Hôpital's rule to take the derivative of the top and bottom, giving:
Plugging in 0 for x gives 1 here.
Example 2
Find
First, you need to rewrite the function into an indeterminate limit fraction:
Now it's indeterminate. Take the derivative of the top and bottom:
Plugging in 0 for x once again gives one.
Example 3
Find
This time, plugging in for x gives you
. You know the drill:
This time, though, there is no x term left! is the answer.
Example 4
Sometimes, forms exist where it is not intuitively obvious how to solve them. One might think the value However, as was noted in the definition of an indeterminate form, this isn't possible to evaluate using the rules learned before now, and we need to use L'Hôpital's rule to solve.
Find
Plugging the value of x into the limit yields
(indeterminate form).
Let
-
= = = (indeterminate form)
We now apply L'Hôpital's rule by taking the derivative of the top and bottom with respect to x.
Returning to the expression above
-
= = (indeterminate form)
We apply L'Hôpital's rule once again
Therefore
And
Careful: this does not prove that because
Exercises
Evaluate the following limits using L'Hôpital's rule:





<h1> 3.11 Extrema and Points of Inflection</h1>
Maxima and minima are points where a function reaches a highest or lowest value, respectively. There are two kinds of extrema (a word meaning maximum or minimum): global and local, sometimes referred to as "absolute" and "relative", respectively. A global maximum is a point that takes the largest value on the entire range of the function, while a global minimum is the point that takes the smallest value on the range of the function. On the other hand, local extrema are the largest or smallest values of the function in the immediate vicinity.
All extrema look like the crest of a hill or the bottom of a bowl on a graph of the function. A global extremum is always a local extremum too, because it is the largest or smallest value on the entire range of the function, and therefore also its vicinity. It is also possible to have a function with no extrema, global or local: y=x is a simple example.
At any extremum, the slope of the graph is necessarily zero, as the graph must stop rising or falling at an extremum, and begin to head in the opposite direction. Because of this, extrema are also commonly called stationary points or turning points. Therefore, the first derivative of a function is equal to zero at extrema. If the graph has one or more of these stationary points, these may be found by setting the first derivative equal to zero and finding the roots of the resulting equation.
However, a slope of zero does not guarantee a maximum or minimum: there is a third class of stationary point called a point of inflexion (also spelled point of inflection). Consider the function
-
.
The derivative is
The slope at x=0 is 0. We have a slope of zero, but while this makes it a stationary point, this doesn't mean that it is a maximum or minimum. Looking at the graph of the function you will see that x=0 is neither, it's just a spot at which the function flattens out. True extrema require a sign change in the first derivative. This makes sense - you have to rise (positive slope) to and fall (negative slope) from a maximum. In between rising and falling, on a smooth curve, there will be a point of zero slope - the maximum. A minimum would exhibit similar properties, just in reverse.
This leads to a simple method to classify a stationary point - plug x values slightly left and right into the derivative of the function. If the results have opposite signs then it is a true maximum/minimum. You can also use these slopes to figure out if it is a maximum or a minimum: the left side slope will be positive for a maximum and negative for a minimum. However, you must exercise caution with this method, as, if you pick a point too far from the extremum, you could take it on the far side of another extremum and incorrectly classify the point.
The Extremum Test
A more rigorous method to classify a stationary point is called the extremum test, or 2nd Derivative Test. As we mentioned before, the sign of the first derivative must change for a stationary point to be a true extremum. Now, the second derivative of the function tells us the rate of change of the first derivative. It therefore follows that if the second derivative is positive at the stationary point, then the gradient is increasing. The fact that it is a stationary point in the first place means that this can only be a minimum. Conversely, if the second derivative is negative at that point, then it is a maximum.
Now, if the second derivative is zero, we have a problem. It could be a point of inflexion, or it could still be an extremum. Examples of each of these cases are below - all have a second derivative equal to zero at the stationary point in question:
has a point of inflexion at
has a minimum at
has a maximum at
However, this is not an insoluble problem. What we must do is continue to differentiate until we get, at the (n+1)th derivative, a non-zero result at the stationary point:
If n is odd, then the stationary point is a true extremum. If the (n+1)th derivative is positive, it is a minimum; if the (n+1)th derivative is negative, it is a maximum. If n is even, then the stationary point is a point of inflexion.
As an example, let us consider the function
We now differentiate until we get a non-zero result at the stationary point at x=0 (assume we have already found this point as usual):
Therefore, (n+1) is 4, so n is 3. This is odd, and the fourth derivative is negative, so we have a maximum. Note that none of the methods given can tell you if this is a global extremum or just a local one. To do this, you would have to set the function equal to the height of the extremum and look for other roots.
Critical Points
Critical points are the points where a function's derivative is 0 or not defined. Suppose we are interested in finding the maximum or minimum on given closed interval of a function that is continuous on that interval. The extreme values of the function on that interval will be at one or more of the critical points and/or at one or both of the endpoints. We can prove this by contradiction. Suppose that the function has maximum at a point
in the interval
where the derivative of the function is defined and not
. If the derivative is positive, then
values slightly greater than
will cause the function to increase. Since
is not an endpoint, at least some of these values are in
. But this contradicts the assumption that
is the maximum of
for
in
. Similarly, if the derivative is negative, then
values slightly less than
will cause the function to increase. Since
is not an endpoint, at least some of these values are in
. This contradicts the assumption that
is the maximum of
for
in
. A similar argument could be made for the minimum.
Example 1
Consider the function on the interval
. The unrestricted function
has no maximum or minimum. On the interval
, however, it is obvious that the minimum will be
, which occurs at
and the maximum will be
, which occurs at
. Since there are no critical points (
exists and equals
everywhere), the extreme values must occur at the endpoints.
Example 2
Find the maximum and minimum of the function on the interval
.
- First start by finding the roots of the function derivative:
- Now evaluate the function at all critical points and endpoints to find the extreme values.
- From this we can see that the minimum on the interval is -24 when
and the maximum on the interval is
when
See "Optimization" for a common application of these principles.
<h1> 3.12 Newton's Method</h1>
Newton's Method (also called the Newton-Raphson method) is a recursive algorithm for approximating the root of a differentiable function. We know simple formulas for finding the roots of linear and quadratic equations, and there are also more complicated formulae for cubic and quartic equations. At one time it was hoped that there would be formulas found for equations of quintic and higher-degree, though it was later shown by Neils Henrik Abel that no such equations exist. The Newton-Raphson method is a method for approximating the roots of polynomial equations of any order. In fact the method works for any equation, polynomial or not, as long as the function is differentiable in a desired interval.
Let As you recursively calculate, the |
In order to explain Newton's method, imagine that is already very close to a zero of
. We know that if we only look at points very close to
then
looks like its tangent line. If
was already close to the place where
was zero, and near
we know that
looks like its tangent line, then we hope the zero of the tangent line at
is a better approximation then
itself.
The equation for the tangent line to at
is given by
Now we set and solve for
.
This value of we feel should be a better guess for the value of
where
We choose to call this value of
, and a little algebra we have
If our intuition was correct and is in fact a better approximation for the root of
, then our logic should apply equally well at
. We could look to the place where the tangent line at
is zero. We call
, following the algebra above we arrive at the formula
And we can continue in this way as long as we wish. At each step, if your current approximation is our new approximation will be
Examples
Find the root of the function.
As you can see is gradually approaching zero (which we know is the root of
). One can approach the function's root with arbitrary accuracy.
Answer:has a root at
.
Notes
This method fails when . In that case, one should choose a new starting place. Occasionally it may happen that
and
have a common root. To detect whether this is true, we should first find the solutions of
, and then check the value of
at these places.
Newton's method also may not converge for every function, take as an example:
For this function choosing any then
would cause successive approximations to alternate back and forth, so no amount of iteration would get us any closer to the root than our first guess.
Newton's method may also fail to converge on a root if the function has a local maximum or minimum that does not cross the x-axis. As an example, consider with initial guess
. In this case, Newton's method will be fooled by the function, which dips toward the x-axis but never crosses it in the vicinity of the initial guess.
See also
- Wikipedia:Newton's method
- Wikibooks:Fractals/Mathematics/Newton_method
- Wikipedia:Abel–Ruffini theorem
<h1> 3.13 Related Rates</h1>
Introduction
One useful application of derivatives is as an aid in the calculation of related rates. What is a related rate? In each case in the following examples the related rate we are calculating is a derivative with respect to some value. We compute this derivative from a rate at which some other known quantity is changing. Given the rate at which something is changing, we are asked to find the rate at which a value related to the rate we are given is changing.
Process for solving related rates problems:
- Write out any relevant formulas and information.
- Take the derivative of the primary equation with respect to time.
- Solve for the desired variable.
- Plug-in known information and simplify.
Notation
Newton's dot notation is used to show the derivative of a variable with respect to time. That is, if is a quantity that depends on time, then
, where
represents the time. This notation is a useful abbreviation in situations where time derivatives are often used, as is the case with related rates.
Examples
Example 1:
- Write out any relevant formulas or pieces of information.
- Take the derivative of the equation above with respect to time. Remember to use the Chain Rule and the Product Rule.

Example 2:
- Write out any relevant formulas and pieces of information.
- Take the derivative of both sides of the volume equation with respect to time.
-
= =
- Solve for
.
- Plug-in known information.

Example 3:
Note: Because the vertical distance is downward in nature, the rate of change of y is negative. Similarly, the horizontal distance is decreasing, therefore it is negative (it is getting closer and closer).
The easiest way to describe the horizontal and vertical relationships of the plane's motion is the Pythagorean Theorem.
- Write out any relevant formulas and pieces of information.
(where s is the distance between the plane and the house)
- Take the derivative of both sides of the distance formula with respect to time.
- Solve for
.
=
- Plug-in known information
-
= = = ft/s

Example 4:
- Write down any relevant formulas and information.
Substitute into the volume equation.
-
= = =
- Take the derivative of the volume equation with respect to time.
- Solve for
.
- Plug-in known information and simplify.
-
= = ft/min

Example 5:
- Write out any relevant formulas and information.
Use the Pythagorean Theorem to describe the motion of the ladder.
(where l is the length of the ladder)
- Take the derivative of the equation with respect to time.
(
is constant so
.)
- Solve for
.
- Plug-in known information and simplify.
-
= = ft/sec

Exercises










<h1> 3.14 Optimization</h1>
Introduction
Optimization is one of the uses of calculus in the real world. Let us assume we are a pizza parlor and wish to maximize profit. Perhaps we have a flat piece of cardboard and we need to make a box with the greatest volume. How does one go about this process?
Obviously, this requires the use of maximums and minimums. We know that we find maximums and minimums via derivatives. Therefore, one can conclude that calculus will be a useful tool for maximizing or minimizing (collectively known as "optimizing") a situation.
Examples
Volume Example
A box manufacturer desires to create a box with a surface area of 100 inches squared. What is the maximum size volume that can be formed by bending this material into a box? The box is to be closed. The box is to have a square base, square top, and rectangular sides.
- Write out known formulas and information
- Eliminate the variable h in the volume equation
-
= =
- Find the derivative of the volume equation in order to maximize the volume
- Set
and solve for
- Plug-in the x value into the volume equation and simplify
-
= =
Answer:![]()
Volume Example II
It is desired to make an open-top box of greatest possible volume from a square piece of tin whose side is , by cutting equal squares out of the corners and then folding up the tin to form the sides. What should be the length of a side of the squares cut out?
If we call the side length of the cut out squares , then each side of the base of the folded box is
, and the height is
. Therefore, the volume function is
.
We must optimize the volume by taking the derivative of the volume function and setting it equal to 0. Since it does not change, is treated as a constant, not a variable.
We can now use the quadratic formula to solve for :
We reject , since it is a minimum (it results in the base length
being 0, making the volume 0). Therefore, the answer is
.
Sales Example
A small retailer can sell n units of a product for a revenue of and at a cost of
, with all amounts in thousands. How many units does it sell to maximize its profit?
The retailer's profit is defined by the equation , which is the revenue generated less the cost. The question asks for the maximum amount of profit which is the maximum of the above equation. As previously discussed, the maxima and minima of a graph are found when the slope of said graph is equal to zero. To find the slope one finds the derivative of
. By using the subtraction rule
:
![]() |
= ![]() |
![]() |
= ![]() |
= ![]() |
Therefore, when the profit will be maximized or minimized. Use the quadratic formula to find the roots, giving {3.798,0.869}. To find which of these is the maximum and minimum the function can be tested:
Because we only consider the functions for all (i.e., you can't have
units), the only points that can be minima or maxima are those two listed above. To show that 3.798 is in fact a maximum (and that the function doesn't remain constant past this point) check if the sign of
changes at this point. It does, and for n greater than 3.798
the value will remain decreasing. Finally, this shows that for this retailer selling 3.798 units would return a profit of $8,588.02.
<h1> 3.15 Euler's Method</h1>
Euler's Method is a method for estimating the value of a function based upon the values of that function's first derivative.
The general algorithm for finding a value of is:
where f is y'(x). In other words, the new value, , is the sum of the old value
and the step size
times the change,
.
You can think of the algorithm as a person traveling with a map: Now I am standing here and based on these surroundings I go that way 1 km. Then, I check the map again and determine my direction again and go 1 km that way. I repeat this until I have finished my trip.
The Euler method is mostly used to solve differential equations of the form

Examples
A simple example is to solve the equation:

This yields and hence, the updating rule is:

Step size = 0.1 is used here.
The easiest way to keep track of the successive values generated by the algorithm is to draw a table with columns for .
The above equation can be e.g. a population model, where y is the population size and x a decease that is reducing the population.
<h1> 3.16 Applications of Derivatives Cumulative Exercises</h1>
Relative Extrema
Find the relative maximum(s) and minimum(s), if any, of the following functions.






Range of Function

Absolute Extrema
Determine the absolute maximum and minimum of the following functions on the given domain

![[0,3]](../../../upload.wikimedia.org/math/e/d/9/ed9c05fe24c0f49f5d73f494a921e0c4.png)

![[-\frac{1}{2},2]](../../../upload.wikimedia.org/math/6/c/1/6c1151423823bf7287ccb7f52b600ce3.png)
Determine Intervals of Change
Find the intervals where the following functions are increasing or decreasing






Determine Intervals of Concavity
Find the intervals where the following functions are concave up or concave down






Word Problems





Graphing Functions
For each of the following, graph a function that abides by the provided characteristics


![f \mbox{ has domain } [-1,1], \; f(-1) = -1, \; f(-\frac{1}{2}) = -2,\; f'(-\frac{1}{2}) = 0,\; f''(x)>0 \mbox{ on } (-1,1)](../../../upload.wikimedia.org/math/2/b/f/2bf8fec6f7f6dd772830f2e98a9ec5c7.png)
Integration
Basics of Integration
<h1> 4.1 Definite Integral</h1>
Suppose we are given a function and would like to determine the area underneath its graph over an interval. We could guess, but how could we figure out the exact area? Below, using a few clever ideas, we actually define such an area and show that by using what is called the definite integral we can indeed determine the exact area underneath a curve.
Definition of the Definite Integral
The rough idea of defining the area under the graph of is to approximate this area with a finite number of rectangles. Since we can easily work out the area of the rectangles, we get an estimate of the area under the graph. If we use a larger number of smaller-sized rectangles we expect greater accuracy with respect to the area under the curve and hence a better approximation. Somehow, it seems that we could use our old friend from differentiation, the limit, and "approach" an infinite number of rectangles to get the exact area. Let's look at such an idea more closely.
Suppose we have a function that is positive on the interval
and we want to find the area
under
between
and
. Let's pick an integer
and divide the interval into
subintervals of equal width (see Figure 1). As the interval
has width
, each subinterval has width
We denote the endpoints of the subintervals by
. This gives us
Now for each pick a sample point
in the interval
and consider the rectangle of height
and width
(see Figure 2). The area of this rectangle is
. By adding up the area of all the rectangles for
we get that the area
is approximated by
A more convenient way to write this is with summation notation:
For each number we get a different approximation. As
gets larger the width of the rectangles gets smaller which yields a better approximation (see Figure 3). In the limit of
as
tends to infinity we get the area
.
Definition of the Definite Integral
Suppose is a continuous function on
and
. Then the definite integral of
between
and
is

![[x_{i-1},x_{i}]](../../../upload.wikimedia.org/math/c/2/c/c2ce9dbbc238ade03ca1098087b24211.png)


It is a fact that if is continuous on
then this limit always exists and does not depend on the choice of the points
. For instance they may be evenly spaced, or distributed ambiguously throughout the interval. The proof of this is technical and is beyond the scope of this section.
Notation







![[a,b]](../../../upload.wikimedia.org/math/2/c/3/2c3d331bc98b44e71cb2aae9edadca7e.png)


One important feature of this definition is that we also allow functions which take negative values. If for all
then
so
. So the definite integral of
will be strictly negative. More generally if
takes on both positive an negative values then
will be the area under the positive part of the graph of
minus the area above the graph of the negative part of the graph (see Figure 4). For this reason we say that
is the signed area under the graph.
Independence of Variable
It is important to notice that the variable did not play an important role in the definition of the integral. In fact we can replace it with any other letter, so the following are all equal:
Each of these is the signed area under the graph of between
and
. Such a variable is often referred to as a dummy variable or a bound variable.
Left and Right Handed Riemann Sums
The following methods are sometimes referred to as L-RAM and R-RAM, RAM standing for "Rectangular Approximation Method."
We could have decided to choose all our sample points to be on the right hand side of the interval
(see Figure 5). Then
for all
and the approximation that we called
for the area becomes
This is called the right-handed Riemann sum, and the integral is the limit
Alternatively we could have taken each sample point on the left hand side of the interval. In this case (see Figure 6) and the approximation becomes
Then the integral of is
The key point is that, as long as is continuous, these two definitions give the same answer for the integral.
Examples
Example 1
In this example we will calculate the area under the curve given by the graph of for
between 0 and 1. First we fix an integer
and divide the interval
into
subintervals of equal width. So each subinterval has width
To calculate the integral we will use the right-handed Riemann sum. (We could have used the left-handed sum instead, and this would give the same answer in the end). For the right-handed sum the sample points are
Notice that . Putting this into the formula for the approximation,
Now we use the formula
to get
To calculate the integral of between
and
we take the limit as
tends to infinity,
Example 2
Next we show how to find the integral of the function between
and
. This time the interval
has width
so
Once again we will use the right-handed Riemann sum. So the sample points we choose are
Thus
We have to calculate each piece on the right hand side of this equation. For the first two,
For the third sum we have to use a formula
to get
Putting this together
Taking the limit as tend to infinity gives
Exercises






Basic Properties of the Integral
From the definition of the integral we can deduce some basic properties. For all the following rules, suppose that f and g are continuous on [a,b].
The Constant Rule
Constant Rule

When f is positive, the height of the function cf at a point x is c times the height of the function f. So the area under cf between a and b is c times the area under f. We can also give a proof using the definition of the integral, using the constant rule for limits,
Example
We saw in the previous section that
.
Using the constant rule we can use this to calculate that
Example
We saw in the previous section that
We can use this and the constant rule to calculate that
There is a special case of this rule used for integrating constants:
Integrating Constants

When and
this integral is the area of a rectangle of height c and width b-a which equals c(b-a).
Example
The addition and subtraction rule
Addition and Subtraction Rules of Integration

As with the constant rule, the addition rule follows from the addition rule for limits:
-
= = =
The subtraction rule can be proved in a similar way.
Example
From above and
so
Example
Exercise




The Comparison Rule
Comparison Rule
- Suppose
for all x in [a,b]. Then
- Suppose
for all x in [a,b]. Then
- Suppose
for all x in [a,b]. Then
If then each of the rectangles in the Riemann sum to calculate the integral of f will be above the y axis, so the area will be non-negative. If
then
and by the first property we get the second property. Finally if
then the area under the graph of f will be greater than the area of rectangle with height m and less than the area of the rectangle with height M (see Figure 7). So
Linearity with respect to endpoints
Additivity with respect to endpoints Suppose . Then
Again suppose that is positive. Then this property should be interpreted as saying that the area under the graph of
between
and
is the area between
and
plus the area between
and
(see Figure 8).
Extension of Additivity with respect to limits of integration
When we have that
so
Also in defining the integral we assumed that . But the definition makes sense even when
, in which case
has changed sign. This gives
With these definitions,

Exercise

Even and odd functions
Recall that a function is called odd if it satisfies
and is called even if
Suppose is a continuous odd function then for any
,
If is a continuous even function then for any
,
Suppose is an odd function and consider first just the integral from
to
. We make the substitution
so
. Notice that if
then
and if
then
. Hence
Now as
is odd,
so the integral becomes
Now we can replace the dummy variable
with any other variable. So we can replace it with the letter
to give
Now we split the integral into two pieces
The proof of the formula for even functions is similar.


<h1> 4.2 Fundamental Theorem of Calculus</h1>
The fundamental theory of calculus is a critical portion of calculus because it links the concept of a derivative to that of an integral. As a result, we can use our knowledge of derivatives to find the area under the curve, which is often quicker and simpler than using the definition of the integral.
Mean Value Theorem for Integration
We will need the following theorem in the discussion of the Fundamental Theorem of Calculus.
Mean Value Theorem for Integration
Suppose
![[a,b]](../../../upload.wikimedia.org/math/2/c/3/2c3d331bc98b44e71cb2aae9edadca7e.png)


![[a,b]](../../../upload.wikimedia.org/math/2/c/3/2c3d331bc98b44e71cb2aae9edadca7e.png)
Proof of the Mean Value Theorem for Integration
satisfies the requirements of the Extreme Value Theorem, so it has a minimum
and a maximum
in
. Since
and since
for all
in
,
we have
Since is continuous, by the Intermediate Value Theorem there is some
with
in
such that
Fundamental Theorem of Calculus
Statement of the Fundamental Theorem
Suppose that f is continuous on [a,b]. We can define a function F by
Fundamental Theorem of Calculus Part I Suppose f is continuous on [a,b] and F is defined by
Then F is differentiable on (a,b) and for all ,
When we have such functions and
where
for every
in some interval
we say that
is the antiderivative of
on
.
Fundamental Theorem of Calculus Part II Suppose that f is continuous on [a,b] and that F is any antiderivative of f. Then
Note: a minority of mathematicians refer to part one as two and part two as one. All mathematicians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.
Proofs
Proof of Fundamental Theorem of Calculus Part I
Suppose x is in (a,b). Pick so that
is also in (a, b). Then
and
.
Subtracting the two equations gives
Now
so rearranging this we have
According to the Mean Value Theorem for Integration, there exists a c in [x, x + Δx] such that
.
Notice that c depends on . Anyway what we have shown is that
,
and dividing both sides by Δx gives
.
Take the limit as we get the definition of the derivative of F at x so we have
.
To find the other limit, we will use the squeeze theorem. The number c is in the interval [x, x + Δx], so x≤ c ≤ x + Δx. Also, and
. Therefore, according to the squeeze theorem,
.
As f is continuous we have
which completes the proof.
Proof of Fundamental Theorem of Calculus Part II
Define Then by the Fundamental Theorem of Calculus part I we know that
is differentiable on
and for all
So is an antiderivative of
. Since we were assuming that
was also an antiderivative for all
,
.
Let . The Mean Value Theorem applied to
on
with
says that
for some in
. But since
for all
in
,
must equal
for all
in
, i.e. g(x) is constant on
.
This implies there is a constant such that for all
,
,
and as is continuous we see this holds when
and
as well. And putting
gives
Notation for Evaluating Definite Integrals
The second part of the Fundamental Theorem of Calculus gives us a way to calculate definite integrals. Just find an antiderivative of the integrand, and subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound. That is
where . As a convenience, we use the notation
to represent .
Integration of Polynomials
Using the power rule for differentiation we can find a formula for the integral of a power using the Fundamental Theorem of Calculus. Let . We want to find an antiderivative for
. Since the differentiation rule for powers lowers the power by 1 we have that
As long as we can divide by
to get
So the function is an antiderivative of
. If
is not in
then
is continuous on
and, by applying the Fundamental Theorem of Calculus, we can calculate the integral of
to get the following rule.
Power Rule of Integration I



![[a,b]](../../../upload.wikimedia.org/math/2/c/3/2c3d331bc98b44e71cb2aae9edadca7e.png)
Notice that we allow all values of , even negative or fractional. If
then this works even if
includes
.
Power Rule of Integration II


Examples
- To find
we raise the power by 1 and have to divide by 4. So
- The power rule also works for negative powers. For instance
- We can also use the power rule for fractional powers. For instance
- Using linearity the power rule can also be thought of as applying to constants. For example,
.
- Using the linearity rule we can now integrate any polynomial. For example
Exercises



<h1> 4.3 Indefinite Integral</h1>
Definition
Now recall that F is said to be an antiderivative of f if . However, F is not the only antiderivative. We can add any constant to F without changing the derivative. With this, we define the indefinite integral as follows:


The function , the function being integrated, is known as the integrand. Note that the indefinite integral yields a family of functions.
Example
Since the derivative of is
, the general antiderivative of
is
plus a constant. Thus,
Example: Finding antiderivatives
Let's take a look at . How would we go about finding the integral of this function? Recall the rule from differentiation that
In our circumstance, we have:
This is a start! We now know that the function we seek will have a power of 3 in it. How would we get the constant of 6? Well,
Thus, we say that is an antiderivative of
.
Exercises


Indefinite integral identities
Basic Properties of Indefinite Integrals
Constant Rule for indefinite integrals

Sum/Difference Rule for indefinite integrals
Indefinite integrals of Polynomials
Say we are given a function of the form, , and would like to determine the antiderivative of f. Considering that
we have the following rule for indefinite integrals:
Power rule for indefinite integrals


Integral of the Inverse function
To integrate , we should first remember
Therefore, since is the derivative of
we can conclude that

Note that the polynomial integration rule does not apply when the exponent is -1. This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.
Integral of the Exponential function
Since
we see that is its own antiderivative. This allows us to find the integral of an exponential function:

Integral of Sine and Cosine
Recall that
So sin x is an antiderivative of cos x and -cos x is an antiderivative of sin x. Hence we get the following rules for integrating sin x and cos x


We will find how to integrate more complicated trigonometric functions in the chapter on integration techniques.
Example
Suppose we want to integrate the function . An application of the sum rule from above allows us to use the power rule and our rule for integrating
as follows,
Exercises


The Substitution Rule
The substitution rule is a valuable asset in the toolbox of any integration greasemonkey. It is essentially the chain rule (a differentiation technique you should be familiar with) in reverse. First, let's take a look at an example:
Preliminary Example
Suppose we want to find . That is, we want to find a function such that its derivative equals
. Stated yet another way, we want to find an antiderivative of
. Since
differentiates to
, as a first guess we might try the function
. But by the Chain Rule,
Which is almost what we want apart from the fact that there is an extra factor of 2 in front. But this is easily dealt with because we can divide by a constant (in this case 2). So,
Thus, we have discovered a function, , whose derivative is
. That is, F is an antiderivative of
. This gives us
Generalization
In fact, this technique will work for more general integrands. Suppose u is a differentiable function. Then to evaluate we just have to notice that by the Chain Rule
As long as is continuous we have that
Now the right hand side of this equation is just the integral of but with respect to u. If we write u instead of u(x) this becomes
So, for instance, if we have worked out that
General Substitution Rule
Now there was nothing special about using the cosine function in the discussion above, and it could be replaced by any other function. Doing this gives us the substitution rule for indefinite integrals:
Substitution rule for indefinite integrals
Assume u is differentiable with continuous derivative and that f is continuous on the range of u. Then

Notice that it looks like you can "cancel" in the expression to leave just a
. This does not really make any sense because
is not a fraction. But it's a good way to remember the substitution rule.
Examples
The following example shows how powerful a technique substitution can be. At first glance the following integral seems intractable, but after a little simplification, it's possible to tackle using substitution.
Example
We will show that
First, we re-write the integral:
.
Now we preform the following substitution:
Which yields:
.
Exercises



Integration by Parts
Integration by parts is another powerful tool for integration. It was mentioned above that one could consider integration by substitution as an application of the chain rule in reverse. In a similar manner, one may consider integration by parts as the product rule in reverse.
Preliminary Example
General Integration by Parts
Integration by parts for indefinite integrals
Suppose f and g are differentiable and their derivatives are continuous. Then
If we write u=f(x) and v=g(x), then by using the Leibniz notation du=f'(x) dx and dv=g'(x) dx the integration by parts rule becomes

Examples
Example
Find
Here we let:
, so that
,
, so that
.
Then:
Example
Find
In this example we will have to use integration by parts twice.
Here we let
, so that
,
, so that
.
Then:
Now to calculate the last integral we use integration by parts again. Let
, so that
,
, so that
and integrating by parts gives
So, finally we obtain
Example
Find
The trick here is to write this integral as
Now let
so
,
so
.
Then using integration by parts,
Example
Find
Again the trick here is to write the integrand as . Then let
- u = arctan(x); du = 1/(1+x2) dx
- v = x; dv = 1·dx
so using integration by parts,
Example
Find
This example uses integration by parts twice. First let,
- u = ex; thus du = exdx
- dv = cos(x)dx; thus v = sin(x)
so
Now, to evaluate the remaining integral, we use integration by parts again, with
- u = ex; du = exdx
- v = -cos(x); dv = sin(x)dx
Then
Putting these together, we have
Notice that the same integral shows up on both sides of this equation, but with opposite signs. The integral does not cancel; it doubles when we add the integral to both sides to get
Exercises




<h1> Failed to match page to section number. Check your argument; if correct, consider updating Template:Calculus/map page. Improper integrals</h1>
The definition of a definite integral:
requires the interval [a,b] be finite. The Fundamental Theorem of Calculus requires that f be continuous on [a,b]. In this section, you will be studying a method of evaluating integrals that fail these requirements—either because their limits of integration are infinite, or because a finite number of discontinuities exist on the interval [a,b]. Integrals that fail either of these requirements are improper integrals. (If you are not familiar with L'Hôpital's rule, it is a good idea to review it before reading this section.)
Improper Integrals with Infinite Limits of Integration
Consider the integral
Assigning a finite upper bound in place of infinity gives
This improper integral can be interpreted as the area of the unbounded region between ,
(the
-axis), and
.
Definition
1. Suppose
exists for all
. Then we define
=
as long as this limit exists and is finite.
If it does exist we say the integral is convergent and otherwise we say it is divergent.
2. Similarly if
exists for all
we define
=
3. Finally suppose
is a fixed real number and that
and
are both convergent. Then we define
=
Example: Convergent Improper Integral
We claim that To do this we calculate
|
Example: Divergent Improper Integral
We claim that the integral
This follows as
Therefore
|
Example: Improper Integral
Find To calculate the integral use integration by parts twice to get Now |
Example: Powers
Show If
Notice that we had to assume that |
Improper Integrals with a Finite Number Discontinuities
First we give a definition for the integral of functions which have a discontinuity at one point.
Definition of improper integrals with a single discontinuity
If f is continuous on the interval [a,b) and is discontinuous at b, we define :
=
If the limit in question exists we say the integral converges and otherwise we say it diverges.
Similarly if f is continuous on the interval(a,b] and is discontinuous at a, we define
=
Finally suppose f has an discontinuity at a point c in (a,b) and is continuous at all other points in [a,b]. If
and
converge we define
=
Example 1
Show If
Notice that we had to assume that which diverges. |
Example 2
The integral which is not correct. In fact the integral diverges since and |
We can also give a definition of the integral of a function with a finite number of discontinuities.
Definition: Improper integrals with finite number of discontinuities
Suppose f is continuous on [a,b] except at points
in [a,b]. We define
as long as each integral on the right converges.
Notice that by combining this definition with the definition for improper integrals with infinite endpoints, we can define the integral of a function with a finite number of discontinuities with one or more infinite endpoints.
Comparison Test
There are integrals which cannot easily be evaluated. However it may still be possible to show they are convergent by comparing them to an integral we already know converges.
Theorem (Comparison Test) Let f and g be continuous functions defined for all
.
- Suppose
for all
. Then if
converges so does
- Suppose
for all
. Then if
diverges so does
A similar theorem holds for improper integrals of the form and for improper integrals with discontinuities.
Example: Use of comparsion test to show convergence
Show that For all x we know that
We have seen that |
Example: Use of Comparsion Test to show divergence
Show that Just as in the previous example we know that We have seen that |
An extension of the comparison theorem
To apply the comparison theorem you do not really need for all
. What we actually need is this inequality holds for sufficiently large x (i.e. there is a number c such that
for all
). For then
so the first integral converges if and only if third does, and we can apply the comparison theorem to the piece.
Example
Show that The reason that this integral converges is because for large x the is the wrong way around to show convergence. Instead we rewrite the integrand as Since the limit Since the integral |
Integration Techniques
4.6 Derivative Rules and the Substitution Rule
4.8 Trigonometric Substitutions
4.10 Rational Functions by Partial Fraction Decomposition
4.11 Tangent Half Angle Substitution
<h1> 4.5 Infinite Sums</h1>
The most basic, and arguably the most difficult, type of evaluation is to use the formal definition of a Riemann integral.
Exact Integrals as Limits of Sums
Using the definition of an integral, we can evaluate the limit as goes to infinity. This technique requires a fairly high degree of familiarity with summation identities. This technique is often referred to as evaluation "by definition," and can be used to find definite integrals, as long as the integrands are fairly simple. We start with definition of the integral:
-
Then picking to be
we get,
In some simple cases, this expression can be reduced to a real number, which can be interpreted as the area under the curve if f(x) is positive on [a,b].
Example 1
Find by writing the integral as a limit of Riemann sums.
In other cases, it is even possible to evaluate indefinite integrals using the formal definition. We can define the indefinite integral as follows:
Example 2
Suppose , then we can evaluate the indefinite integral as follows.
<h1> 4.6 Derivative Rules and the Substitution Rule</h1>
After learning a simple list of antiderivatives, it is time to move on to more complex integrands, which are not at first readily integrable. In these first steps, we notice certain special case integrands which can be easily integrated in a few steps.
Recognizing Derivatives and Reversing Derivative Rules
If we recognize a function as being the derivative of a function
, then we can easily express the antiderivative of
:
For example, since
we can conclude that
Similarly, since we know is its own derivative,
The power rule for derivatives can be reversed to give us a way to handle integrals of powers of . Since
,
we can conclude that
or, a little more usefully,
.
Integration by Substitution
For many integrals, a substitution can be used to transform the integrand and make possible the finding of an antiderivative. There are a variety of such substitutions, each depending on the form of the integrand.
The objective of Integration by substitution is to substitute the integrand from an expression with variable x to an expression with variable where
Theory
We want to transform the Integral from a function of x to a function of u
Starting with
Steps
![]() |
![]() |
(1) | ie ![]() |
|
![]() |
(2) | ie ![]() |
||
![]() |
(3) | ie ![]() |
||
![]() |
(4) | ie Now equate ![]() ![]() |
||
![]() |
(5) | ie ![]() |
||
![]() |
(6) | ie ![]() |
||
![]() |
(7) | ie We have achieved our desired result |
Procedure
- Calculate
- Calculate
which is
and make sure you express the result in terms of the variable u
- Calculate
- Calculate
Integrating with the derivative present
If a component of the integrand can be viewed as the derivative of another component of the integrand, a substitution can be made to simplify the integrand.
For example, in the integral
we see that is the derivative of
. Letting
we have
or, in order to apply it to the integral,
.
With this we may write
Note that it was not necessary that we had exactly the derivative of in our integrand. It would have been sufficient to have any constant multiple of the derivative.
For instance, to treat the integral
we may let . Then
and so
the right-hand side of which is a factor of our integrand. Thus,
In general, the integral of a power of a function times that function's derivative may be integrated in this way. Since ,
we have
Therefore, ![]() |
![]() |
![]() |
|
![]() |
There is a similar rule for definite integrals, but we have to change the endpoints.
Substitution rule for definite integrals
Assume u is differentiable with continuous derivative and that f is continuous on the range of u. Suppose


Examples
Consider the integral
By using the substitution u = x2 + 1, we obtain du = 2x dx and
Note how the lower limit x = 0 was transformed into u = 02 + 1 = 1 and the upper limit x = 2 into u = 22 + 1 = 5.
Proof of the substitution rule
We will now prove the substitution rule for definite integrals. Let H be an anti derivative of h so
.
Suppose we have a differentiable function, such that
, and numbers
and
derived from some given numbers,
and
.
By the Fundamental Theorem of Calculus, we have
Next we define a function by the rule
Naturally
Then by the Chain rule F is differentiable with derivative
Integrating both sides with respect to and using the Fundamental Theorem of Calculus we get
But by the definition of this equals
Hence
which is the substitution rule for definite integrals.
Exercises
Evaluate the following using a suitable substitution.






<h1> 4.7 Integration by Parts</h1>
Continuing on the path of reversing derivative rules in order to make them useful for integration, we reverse the product rule.
Integration by parts
If where
and
are functions of
, then
Rearranging,
Therefore,
Therefore,
, or
This is the integration by parts formula. It is very useful in many integrals involving products of functions, as well as others.
For instance, to treat
we choose and
. With these choices, we have
and
, and we have
Note that the choice of and
was critical. Had we chosen the reverse, so that
and
, the result would have been
The resulting integral is no easier to work with than the original; we might say that this application of integration by parts took us in the wrong direction.
So the choice is important. One general guideline to help us make that choice is, if possible, to choose to be the factor of the integrand which becomes simpler when we differentiate it. In the last example, we see that
does not become simpler when we differentiate it:
is no simpler than
.
An important feature of the integration by parts method is that we often need to apply it more than once. For instance, to integrate
,
we start by choosing and
to get
Note that we still have an integral to take care of, and we do this by applying integration by parts again, with and
, which gives us
So, two applications of integration by parts were necessary, owing to the power of in the integrand.
Note that any power of x does become simpler when we differentiate it, so when we see an integral of the form
one of our first thoughts ought to be to consider using integration by parts with . Of course, in order for it to work, we need to be able to write down an antiderivative for
.
Example
Use integration by parts to evaluate the integral
Solution: If we let and
, then we have
and
. Using our rule for integration by parts gives
We do not seem to have made much progress. But if we integrate by parts again with and
and hence
and
, we obtain
We may solve this identity to find the anti-derivative of and obtain
With definite integral
For definite integrals the rule is essentially the same, as long as we keep the endpoints.
Integration by parts for definite integrals Suppose f and g are differentiable and their derivatives are continuous. Then
.
This can also be expressed in Leibniz notation.
More Examples
Examples Set 1: Integration by Parts
Exercises
Evaluate the following using integration by parts.





<h1> 4.8 Trigonometric Substitutions</h1>
The idea behind the trigonometric substitution is quite simple: to replace expressions involving square roots with expressions that involve standard trigonometric functions, but no square roots. Integrals involving trigonometric functions are often easier to solve than integrals involving square roots.
Let us demonstrate this idea in practice. Consider the expression . Probably the most basic trigonometric identity is
for an arbitrary angle
. If we replace
in this expression by
, with the help of this trigonometric identity we see
Note that we could write since we replaced
with
.
We would like to mention that technically one should write the absolute value of , in other words
as our final answer since
for all possible
. But as long as we are careful about the domain of all possible
and how
is used in the final computation, omitting the absolute value signs does not constitute a problem. However, we cannot directly interchange the simple expression
with the complicated
wherever it may appear, we must remember when integrating by substitution we need to take the derivative into account. That is we need to remember that
, and to get a integral that only involves
we need to also replace
by something in terms of
. Thus, if we see an integral of the form
we can rewrite it as
Notice in the expression on the left that the first comes from replacing the
and the
comes from substituting for the
.
Since our original integral reduces to:
.
These last two integrals are easily handled. For the first integral we get
For the second integral we do a substitution, namely (and
) to get:
Finally we see that:
However, this is in terms of and not in terms of
, so we must substitute back in order to rewrite the answer in terms of
.
That is we worked out that:
So we arrive at our final answer
As you can see, even for a fairly harmless looking integral this technique can involve quite a lot of calculation. Often it is helpful to see if a simpler method will suffice before turning to trigonometric substitution. On the other hand, frequently in the case of integrands involving square roots, this is the most tractable way to solve the problem. We begin with giving some rules of thumb to help you decide which trigonometric substitutions might be helpful.
If the integrand contains a single factor of one of the forms we can try a trigonometric substitution.
- If the integrand contains
let
and use the identity
- If the integrand contains
let
and use the identity
- If the integrand contains
let
and use the identity
Sine substitution

If the integrand contains a piece of the form we use the substitution
This will transform the integrand to a trigonometric function. If the new integrand can't be integrated on sight then the tan-half-angle substitution described below will generally transform it into a more tractable algebraic integrand.
E.g., if the integrand is √(1-x2),
If the integrand is √(1+x)/√(1-x), we can rewrite it as
Then we can make the substitution
Tangent substitution

When the integrand contains a piece of the form we use the substitution
E.g., if the integrand is (x2+a2)-3/2 then on making this substitution we find
If the integral is
then on making this substitution we find
After integrating by parts, and using trigonometric identities, we've ended up with an expression involving the original integral. In cases like this we must now rearrange the equation so that the original integral is on one side only
As we would expect from the integrand, this is approximately z2/2 for large z.
In some cases it is possible to do trigonometric substitution in cases when there is no appearing in the integral.
Example
The denominator of this function is equal to . This suggests that we try to substitute
and use the identity
. With this substitution, we obtain that
and thus
Using the initial substitution gives
Secant substitution

If the integrand contains a factor of the form we use the substitution
Example 1
Find
Example 2
Find
We can now integrate by parts
Exercise
Evaluate the following using an appropriate trigonometric substitution.

<h1> 4.9 Trigonometric Integrals</h1>
When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.
Powers of Sine and Cosine
We will give a general method to solve generally integrands of the form . First let us work through an example.
Notice that the integrand contains an odd power of cos. So rewrite it as
We can solve this by making the substitution so
. Then we can write the whole integrand in terms of
by using the identity
.
So
This method works whenever there is an odd power of sine or cosine.
To evaluate
when either
or
is odd.
- If
is odd substitute
and use the identity
.
- If
is odd substitute
and use the identity
.
Example
Find .
As there is an odd power of we let
so
. Notice that when
we have
and when
we have
.
When both and
are even things get a little more complicated.
To evaluate
when both
and
are even.
Use the identitiesand
.
Example
Find
As and
we have
and expanding, the integrand becomes
Using the multiple angle identities
then we obtain on evaluating
Powers of Tan and Secant
To evaluate
.
- If
is even and
then substitute
and use the identity
.
- If
and
are both odd then substitute
and use the identity
.
- If
is odd and
is even then use the identity
and apply a reduction formula to integrate
, using the examples below to integrate when
.
Example 1
Find .
There is an even power of . Substituting
gives
so
Example 2
Find .
Let so
. Then
Example 3
Find .
The trick to do this is to multiply and divide by the same thing like this:
Making the substitution so
More trigonometric combinations
For the integrals
or
or
use the identities
Example 1
Find
We can use the fact that , so
Now use the oddness property of to simplify
And now we can integrate
Example 2
Find:.
Using the identities
Then
<h1> 4.10 Rational Functions by Partial Fractional Decomposition</h1>
Suppose we want to find . One way to do this is to simplify the integrand by finding constants
and
so that
This can be done by cross multiplying the fraction which gives
As both sides have the same denominator we must have
This is an equation for so it must hold whatever value
is. If we put in
we get
and putting
gives
so
. So we see that
Returning to the original integral
-
= =
Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initial integral to a sum of simpler integrals. In fact this method works to integrate any rational function.
Method of Partial Fractions
To decompose the rational function
:
- Step 1 Use long division to ensure that the degree of
is less than the degree of
(see Breaking up a rational function in section
1.1).
- Step 2 Factor Q(x) as far as possible.
- Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.
To factor Q(x) we have to write it as a product of linear factors (of the form ) and irreducible quadratic factors (of the form
with
).
Some of the factors could be repeated. For instance if we factor
as
It is important that in each quadratic factor we have , otherwise it is possible to factor that quadratic piece further. For example if
then we can write
We will now show how to write as a sum of terms of the form
and
Exactly how to do this depends on the factorization of and we now give four cases that can occur.
Q(x) is a product of linear factors with no repeats
This means that where no factor is repeated and no factor is a multiple of another.
For each linear term we write down something of the form , so in total we write
Example 1
Find Here we have Multiply both sides by the denominator Substitute in three values of x to get three equations for the unknown constants, so We can now integrate the left hand side. |
Exercises
Evaluate the following by the method partial fraction decomposition.


Q(x) is a product of linear factors some of which are repeated
If appears in the factorisation of
k-times then instead of writing the piece
we use the more complicated expression
Example 2
Find Here Multiply both sides by the denominator Substitute in three values of so We can now integrate the left hand side. |
Exercise

Q(x) contains some quadratic pieces which are not repeated
If appears we use
Exercises
Evaluate the following using the method of partial fractions.


Q(x) contains some repeated quadratic factors
If appears k-times then use
Exercise
Evaluate the following using the method of partial fractions.

<h1> 4.11 Tangent Half Angle Substitution</h1>
Another useful change of variables is the Weierstrass substitution, named after Karl Weierstrass:
With this transformation, using the double-angle trigonometric identities,
This transforms a trigonometric integral into a algebraic integral, which may be easier to integrate.
For example, if the integrand is 1/(1 + sin x) then
This method can be used to further simplify trigonometric integrals produced by the changes of variables described earlier.
For example, if we are considering the integral
we can first use the substitution x = sin θ, which gives
then use the tan-half-angle substition to obtain
In effect, we've removed the square root from the original integrand. We could do this with a single change of variables, but doing it in two steps gives us the opportunity of doing the trigonometric integral another way.
Having done this, we can split the new integrand into partial fractions, and integrate.
This result can be further simplified by use of the identities
ultimately leading to
In principle, this approach will work with any integrand which is the square root of a quadratic multiplied by the ratio of two polynomials. However, it should not be applied automatically.
E.g., in this last example, once we deduced
we could have used the double angle formula, since this contains only even powers of cos and sin. Doing that gives
Using tan-half-angle on this new, simpler, integrand gives
This can be integrated on sight to give
This is the same result as before, but obtained with less algebra, which shows why it is best to look for the most straightforward methods at every stage.
A more direct way of evaluating the integral I is to substitute t = tan θ right from the start, which will directly bring us to the line
above. More generally, the substitution t = tan x gives us
so this substitution is the preferable one to use if the integrand is such that all the square roots would disappear after substitution, as is the case in the above integral.
Example
Using the trigonometric substitution , then
and
(when
). So,
Alternate Method
In general, to evaluate integrals of the form
it is extremely tedious to use the aforementioned "tan half angle" substitution directly, as one easily ends up with a rational function with a 4th degree denominator. Instead, we may first write the numerator as
Then the integral can be written as
which can be evaluated much more easily.
Example
Evaluate
Let
Then
Comparing coefficients of cos x, sin x and the constants on both sides, we obtain
yielding p = q = 1/2, r = 2. Substituting back into the integrand,
The last integral can now be evaluated using the "tan half angle" substitution described above, and we obtain
The original integral is thus
<h1> 4.12 Reduction Formula</h1>
A reduction formula is one that enables us to solve an integral problem by reducing it to a problem of solving an easier integral problem, and then reducing that to the problem of solving an easier problem, and so on.
For example, if we let
Integration by parts allows us to simplify this to
which is our desired reduction formula. Note that we stop at
.
Similarly, if we let
then integration by parts lets us simplify this to
Using the trigonometric identity, , we can now write
Rearranging, we get
Note that we stop at or 2 if
is odd or even respectively.
As in these two examples, integrating by parts when the integrand contains a power often results in a reduction formula.
<h1> 4.13 Irrational Functions</h1>
Integration of irrational functions is more difficult than rational functions, and many cannot be done. However, there are some particular types that can be reduced to rational forms by suitable substitutions.
Type 1
Integrand contains
Use the substitution .
Example
Find .
Type 2
Integral is of the form
Write as
.
Example
Find .
Type 3
Integrand contains ,
or
This was discussed in "trigonometric substitutions above". Here is a summary:
- For
, use
.
- For
, use
.
- For
, use
.
Type 4
Integral is of the form
Use the substitution .
Example
Find .
Type 5
Other rational expressions with the irrational function
- If
, we can use
.
- If
, we can use
.
- If
can be factored as
, we can use
.
- If
and
can be factored as
, we can use
<h1> 4.14 Numerical Approximations</h1>
It is often the case, when evaluating definite integrals, that an antiderivative for the integrand cannot be found, or is extremely difficult to find. In some instances, a numerical approximation to the value of the definite value will suffice. The following techniques can be used, and are listed in rough order of ascending complexity.
Riemann Sum
This comes from the definition of an integral. If we pick n to be finite, then we have:
where is any point in the i-th sub-interval
on [a,b].
Right Rectangle
A special case of the Riemann sum, where we let , in other words the point on the far right-side of each sub-interval on, [a,b]. Again if we pick n to be finite, then we have:
Left Rectangle
Another special case of the Riemann sum, this time we let , which is the point on the far left side of each sub-interval on [a,b]. As always, this is an approximation when n is finite. Thus, we have:
Trapezoidal Rule
Simpson's Rule
Remember, n must be even,
Further reading
<h1> 4.15 Integration Exercises</h1>
Integration of Polynomials
Evaluate the following:





Indefinite Integration
Find the general antiderivative of the following:








Integration by parts





Applications of Integration
Area
Introduction
Finding the area between two curves, usually given by two explicit functions, is often useful in calculus.
In general the rule for finding the area between two curves is
or
If f(x) is the upper function and g(x) is the lower function
This is true whether the functions are in the first quadrant or not.
Area between two curves
Suppose we are given two functions y1=f(x) and y2=g(x) and we want to find the area between them on the interval [a,b]. Also assume that f(x)≥ g(x) for all x on the interval [a,b]. Begin by partitioning the interval [a,b] into n equal subintervals each having a length of Δx=(b-a)/n. Next choose any point in each subinterval, xi*. Now we can 'create' rectangles on each interval. At the point xi*, the height of each rectangle is f(xi*)-g(xi*) and the width is Δx. Thus the area of each rectangle is [f(xi*)-g(xi*)]Δx. An approximation of the area, A, between the two curves is
.
Now we take the limit as n approaches infinity and get
which gives the exact area. Recalling the definition of the definite integral we notice that
.
This formula of finding the area between two curves is sometimes known as applying integration with respect to the x-axis since the rectangles used to approximate the area have their bases lying parallel to the x-axis. It will be most useful when the two functions are of the form y1=f(x) and y2=g(x). Sometimes however, one may find it simpler to integrate with respect to the y-axis. This occurs when integrating with respect to the x-axis would result in more than one integral to be evaluated. These functions take the form x1=f(y) and x2=g(y) on the interval [c,d]. Note that [c,d] are values of y. The derivation of this case is completely identical. Similar to before, we will assume that f(y)≥ g(y) for all y on [c,d]. Now, as before we can divide the interval into n subintervals and create rectangles to approximate the area between f(y) and g(y). It may be useful to picture each rectangle having their 'width', Δy, parallel to the y-axis and 'height', f(yi*)-g(yi*) at the point yi*, parallel to the x-axis. Following from the work above we may reason that an approximation of the area, A, between the two curves is
.
As before, we take the limit as n approaches infinity to arrive at
,
which is nothing more than a definite integral, so
.
Regardless of the form of the functions, we basically use the same formula.
Volume
When we think about volume from an intuitive point of view, we typically think of it as the amount of "space" an item occupies. Unfortunately assigning a number that measures this amount of space can prove difficult for all but the simplest geometric shapes. Calculus provides a new tool that can greatly extend our ability to calculate volume. In order to understand the ideas involved it helps to think about the volume of a cylinder. The volume of a cylinder is calculated using the formula . The base of the cylinder is a circle whose area is given by
. Notice that the volume of a cylinder is derived by taking the area of its base and multiplying by the height
. For more complicated shapes, we could think of approximating the volume by taking the area of some cross section at some height
and multiplying by some small change in height
then adding up the heights of all of these approximations from the bottom to the top of the object. This would appear to be a Riemann sum. Keeping this in mind, we can develop a more general formula for the volume of solids in
(3 dimensional space).
Formal Definition
Formally the ideas above suggest that we can calculate the volume of a solid by calculating the integral of the cross-sectional area along some dimension. In the above example of a cylinder, the every cross section was given by the same circle, so the cross-sectional area is therefore a constant function, and the dimension of integration was vertical (although it could have been any one we desired). Generally, if is a solid that lies in
between
and
, let
denote the area of a cross section taken in the plane perpendicular to the x direction, and passing through the point x. If the function
is continuous on
, then the volume
of the solid
is given by:
Examples
Example 1: A right cylinder
Now we will calculate the volume of a right cylinder using our new ideas about how to calculate volume. Since we already know the formula for the volume of a cylinder this will give us a "sanity check" that our formulas make sense. First, we choose a dimension along which to integrate. In this case, it will greatly simplify the calculations to integrate along the height of the cylinder, so this is the direction we will choose. Thus we will call the vertical direction (see Figure 1). Now we find the function,
, which will describe the cross-sectional area of our cylinder at a height of
. The cross-sectional area of a cylinder is simply a circle. Now simply recall that the area of a circle is
, and so
. Before performing the computation, we must choose our bounds of integration. In this case, we simply define
to be the base of the cylinder, and so we will integrate from
to
, where
is the height of the cylinder. Finally, we integrate:
This is exactly the familiar formula for the volume of a cylinder.
Example 2: A right circular cone
For our next example we will look at an example where the cross sectional area is not constant. Consider a right circular cone. Once again the cross sections are simply circles. But now the radius varies from the base of the cone to the tip. Once again we choose to be the vertical direction, with the base at
and the tip at
, and we will let
denote the radius of the base. While we know the cross sections are just circles we cannot calculate the area of the cross sections unless we find some way to determine the radius of the circle at height
.
Luckily in this case it is possible to use some of what we know from geometry. We can imagine cutting the cone perpendicular to the base through some diameter of the circle all the way to the tip of the cone. If we then look at the flat side we just created, we will see simply a triangle, whose geometry we understand well. The right triangle from the tip to the base at height is similar to the right triangle from the tip to the base at height
. This tells us that
. So that we see that the radius of the circle at height
is
. Now using the familiar formula for the area of a circle we see that
.
Now we are ready to integrate.
-
By u-substitution we may let , then
and our integral becomes
Example 3: A sphere
In a similar fashion, we can use our definition to prove the well known formula for the volume of a sphere. First, we must find our cross-sectional area function, . Consider a sphere of radius
which is centered at the origin in
. If we again integrate vertically then
will vary from
to
. In order to find the area of a particular cross section it helps to draw a right triangle whose points lie at the center of the sphere, the center of the circular cross section, and at a point along the circumference of the cross section. As shown in the diagram the side lengths of this triangle will be
,
, and
. Where
is the radius of the circular cross section. Then by the Pythagorean theorem
and find that
. It is slightly helpful to notice that
so we do not need to keep the absolute value.
So we have that
Extension to Non-trivial Solids
Now that we have shown our definition agrees with our prior knowledge, we will see how it can help us extend our horizons to solids whose volumes are not possible to calculate using elementary geometry.
Volume of solids of revolution
In this section we cover solids of revolution and how to calculate their volume. A solid of revolution is a solid formed by revolving a 2-dimensional region around an axis. For example, revolving the semi-circular region bounded by the curve and the line
around the
-axis produces a sphere. There are two main methods of calculating the volume of a solid of revolution using calculus: the disk method and the shell method.
Disk Method
Consider the solid formed by revolving the region bounded by the curve , which is continuous on
, and the lines
,
and
around the
-axis. We could imagine approximating the volume by approximating
with the stepwise function
shown in figure 2, which uses a right-handed approximation to the function. Now when the region is revolved, the region under each step sweeps out a cylinder, whose volume we know how to calculate, i.e.
, where is the radius of the cylinder and
is the cylinder's height. This process is reminiscent of the Riemann process we used to calculate areas earlier. Let's try to write the volume as a Riemann sum and from that equate the volume to an integral by taking the limit as the subdivisions get infinitely small.
Consider the volume of one of the cylinders in the approximation, say the -th one from the left. The cylinder's radius is the height of the step function, and the thickness is the length of the subdivision. With
subdivisions and a length of
for the total length of the region, each subdivision has width
Since we are using a right-handed approximation, the -th sample point will be
So the volume of the -th cylinder is
Summing all of the cylinders in the region from to
, we have
Taking the limit as approaches infinity gives us the the exact volume
, which is equivalent to the integral
Example: Volume of a Sphere
Let's calculate the volume of a sphere using the disk method. Our generating region will be the region bounded by the curve |
Exercises










Washer Method
The washer method is an extension of the disk method to solids of revolution formed by revolving an area bounded between two curves around the -axis. Consider the solid of revolution formed by revolving the region in figure 3 around the
-axis. The curve
is the same as that in figure 1, but now our solid has an irregularly shaped hole through its center whose volume is that of the solid formed by revolving the curve
around the
-axis. Our approximating region has the same upper boundary,
as in figure 2, but now we extend only down to
rather than all the way down to the
-axis. Revolving each block around the
-axis forms a washer-shaped solid with outer radius
and inner radius
. The volume of the
-th hollow cylinder is
where and
. The volume of the entire approximating solid is
Taking the limit as approaches infinity gives the volume
Exercises









Shell Method
The shell method is another technique for finding the volume of a solid of revolution. Using this method sometimes makes it easier to set up and evaluate the integral. Consider the solid of revolution formed by revolving the region in figure 5 around the -axis. While the generating region is the same as in figure 1, the axis of revolution has changed, making the disk method impractical for this problem. However, dividing the region up as we did previously suggests a similar method of finding the volume, only this time instead of adding up the volume of many approximating disks, we will add up the volume of many cylindrical shells. Consider the solid formed by revolving the region in figure 6 around the
-axis. The
-th rectangle sweeps out a hollow cylinder with height
and with inner radius
and outer radius
, where
and
, the volume of which is
The volume of the entire approximating solid is
Taking the limit as approaches infinity gives us the exact volume
Since is continuous on
, the Extreme Value Theorem implies that
has some maximum,
, on
. Using this and the fact that
, we have
But
So by the Squeeze Theorem
, which is just the integral
Exercises







Arc length
Suppose that we are given a function that is continuous on an interval
and we want to calculate the length of the curve drawn out by the graph of
from
to
. If the graph were a straight line this would be easy — the formula for the length of the line is given by Pythagoras' theorem. And if the graph were a piecewise linear function we can calculate the length by adding up the length of each piece.
The problem is that most graphs are not linear. Nevertheless we can estimate the length of the curve by approximating it with straight lines. Suppose the curve is given by the formula
for
. We divide the interval
into
subintervals with equal width
and endpoints
. Now let
so
is the point on the curve above
. The length of the straight line between
and
is
So an estimate of the length of the curve is the sum
As we divide the interval into more pieces this gives a better estimate for the length of
. In fact we make that a definition.
The length of the curve for
is defined to be
The Arclength Formula
Suppose that is continuous on
. Then the length of the curve given by
between
and
is given by
And in Leibniz notation
Proof: Consider By the Mean Value Theorem there is a point
in
such that
So
Putting this into the definition of the length of gives
Now this is the definition of the integral of the function between
and
(notice that
is continuous because we are assuming that
is continuous). Hence
as claimed.
Example: Length of the curve
![]() ![]() ![]() As a sanity check of our formula, let's calculate the length of the "curve" and so the length of the curve, Now let's use the formula |
Exercises






Arclength of a parametric curve
For a parametric curve, that is, a curve defined by and
, the formula is slightly different:
Proof: The proof is analogous to the previous one: Consider and
. By the Mean Value Theorem there are points
and
in
such that
and
So
Putting this into the definition of the length of the curve gives
This is equivalent to:
Exercises










Surface area
Suppose we are given a function f and we want to calculate the surface area of the function f rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.
If the function f is a straight line, other methods such as surface area formulas for cylinders and conical frustra can be used. However, if f is not linear, an integration technique must be used.
Recall the formula for the lateral surface area of a conical frustum:
where r is the average radius and l is the slant height of the frustum.
For y=f(x) and , we divide [a,b] into subintervals with equal width Δx and endpoints
. We map each point
to a conical frustum of width Δx and lateral surface area
.
We can estimate the surface area of revolution with the sum
As we divide [a,b] into smaller and smaller pieces, the estimate gives a better value for the surface area.
Definition (Surface of Revolution)
The surface area of revolution of the curve y=f(x) about a line for is defined to be
The Surface Area Formula
Suppose f is a continuous function on the interval [a,b] and r(x) represents the distance from f(x) to the axis of rotation. Then the lateral surface area of revolution about a line is given by
And in Leibniz notation
Proof:
-
= = =
As and
, we know two things:
1. the average radius of each conical frustum approaches a single value
2. the slant height of each conical frustum equals an infitesmal segment of arc length
From the arc length formula discussed in the previous section, we know that
Therefore
-
= =
Because of the definition of an integral , we can simplify the sigma operation to an integral.
Or if f is in terms of y on the interval [c,d]
Work
Centre of mass
Exercises
See the exercises for Integration
Parametric Equations
Introduction
Introduction
Parametric equations are typically definied by two equations that specify both the x and y coordinates of a graph using a parameter. They are graphed using the parameter (usually t) to figure out both the x and y coordinates.
Example 1:
Note: This parametric equation is equivalent to the rectangular equation .
Example 2:
Note: This parametric equation is equivalent to the rectangular equation and the polar equation
.
Parametric equations can be plotted by using a t-table to show values of x and y for each value of t. They can also be plotted by eliminating the parameter though this method removes the parameter's importance.
Forms of Parametric Equations
Parametric equations can be described in three ways:
- Parametric form
- Vector form
- An equality
The first two forms are used far more often, as they allow us to find the value of the component at the given value of the parameter. The final form is used less often; it allows us to verify a solution to the equation, or find the parameter (or some constant multiple thereof).
Parametric Form
A parametric equation can be shown in parametric form by describing it with a system of equations. For instance:
Vector Form
Vector form can be used to describe a parametric equation in a similar manner to parametric form. In this case, a position vector is given:
Equalities
A parametric equation can also be described with a set of equalities. This is done by solving for the parameter, and equating the components. For example:
From here, we can solve for t:
And hence equate the two right-hand sides:
Converting Parametric Equations
There are a few common place methods used to change a parametric equation to rectangular form. The first involves solving for t in one of the two equations and then replacing the new expression for t with the variable found in the second equation.
Example 1:
becomes
Example 2:
Given
Isolate the trigonometric functions
Use the "Beloved Identity"
Differentiation
Taking Derivatives of Parametric Systems
Just as we are able to differentiate functions of x, we are able to differentiate x and y, which are functions of t. Consider:
We would find the derivative of x with respect to t, and the derivative of y with respect to t:
In general, we say that if
and
then:
and
It's that simple.
This process works for any amount of variables.
Slope of Parametric Equations
In the above process, x' has told us only the rate at which x is changing, not the rate for y, and vice versa. Neither is the slope.
In order to find the slope, we need something of the form .
We can discover a way to do this by simple algebraic manipulation:
So, for the example in section 1, the slope at any time t:
In order to find a vertical tangent line, set the horizontal change, or x', equal to 0 and solve.
In order to find a horizontal tangent line, set the vertical change, or y', equal to 0 and solve.
If there is a time when both x' and y' are 0, that point is called a singular point.
Concavity of Parametric Equations
Solving for the second derivative of a parametric equation can be more complex than it may seem at first glance. When you have take the derivative of in terms of t, you are left with
:
.
By multiplying this expression by , we are able to solve for the second derivative of the parametric equation:
.
Thus, the concavity of a parametric equation can be described as:
So for the example in sections 1 and 2, the concavity at any time t:
Integration
Introduction
Because most parametric equations are given in explicit form, they can be integrated like many other equations. Integration has a variety of applications with respect to parametric equations, especially in kinematics and vector calculus.
So, taking a simple example:
Polar Equations
=Introduction
The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the more familiar Cartesian coordinate system or rectangular coordinate system, such a relationship can only be found through trigonometric formulae.
As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as ) denotes the point's distance from a central point known as the pole (equivalent to the origin in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by θ or
) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or polar axis (which is equivalent to the positive x-axis in the Cartesian coordinate plane).
Plotting points with polar coordinates
Each point in the polar coordinate system can be described with the two polar coordinates, which are usually called (the radial coordinate) and θ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as φ or
). The
coordinate represents the radial distance from the pole, and the θ coordinate represents the anticlockwise (counterclockwise) angle from the 0° ray (sometimes called the polar axis), known as the positive x-axis on the Cartesian coordinate plane.
For example, the polar coordinates (3, 60°) would be plotted as a point 3 units from the pole on the 60° ray. The coordinates (−3, 240°) would also be plotted at this point because a negative radial distance is measured as a positive distance on the opposite ray (the ray reflected about the origin, which differs from the original ray by 180°).
One important aspect of the polar coordinate system, not present in the Cartesian coordinate system, is that a single point can be expressed with an infinite number of different coordinates. This is because any number of multiple revolutions can be made around the central pole without affecting the actual location of the point plotted. In general, the point (, θ) can be represented as (
, θ ±
×360°) or (−
, θ ± (2
+ 1)180°), where
is any integer.
The arbitrary coordinates (0, θ) are conventionally used to represent the pole, as regardless of the θ coordinate, a point with radius 0 will always be on the pole. To get a unique representation of a point, it is usual to limit to negative and non-negative numbers
≥ 0 and θ to the interval [0, 360°) or (−180°, 180°] (or, in radian measure, [0, 2π) or (−π, π]).
Angles in polar notation are generally expressed in either degrees or radians, using the conversion 2π rad = 360°. The choice depends largely on the context. Navigation applications use degree measure, while some physics applications (specifically rotational mechanics) and almost all mathematical literature on calculus use radian measure.
Converting between polar and Cartesian coordinates
The two polar coordinates and θ can be converted to the Cartesian coordinates
and
by using the trigonometric functions sine and cosine:
while the two Cartesian coordinates and
can be converted to polar coordinate
by
(by a simple application of the Pythagorean theorem).
To determine the angular coordinate θ, the following two ideas must be considered:
- For
= 0, θ can be set to any real value.
- For
≠ 0, to get a unique representation for θ, it must be limited to an interval of size 2π. Conventional choices for such an interval are [0, 2π) and (−π, π].
To obtain θ in the interval [0, 2π), the following may be used ( denotes the inverse of the tangent function):
To obtain θ in the interval (−π, π], the following may be used:
One may avoid having to keep track of the numerator and denominator signs by use of the atan2 function, which has separate arguments for the numerator and the denominator.
Polar equations
The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining as a function of θ. The resulting curve then consists of points of the form (
(θ), θ) and can be regarded as the graph of the polar function
.
Different forms of symmetry can be deduced from the equation of a polar function . If
(−θ) =
(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, if
(π−θ) =
(θ) it will be symmetric about the vertical (90°/270°) ray, and if
(θ−α°) =
(θ) it will be rotationally symmetric α° counterclockwise about the pole.
Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.
Circle
The general equation for a circle with a center at (0, φ) and radius
is
This can be simplified in various ways, to conform to more specific cases, such as the equation
for a circle with a center at the pole and radius .
Line
Radial lines (those running through the pole) are represented by the equation
,
where φ is the angle of elevation of the line; that is, φ = arctan where
is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (
0, φ) has the equation
Polar rose
A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
for any constant (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-petaled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a represents the length of the petals of the rose.
Archimedean spiral
The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation
Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the Conic Sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.
Conic sections
A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's semi-major axis lies along the polar axis) is given by:
where e is the eccentricity and is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If e > 1, this equation defines a hyperbola; if e = 1, it defines a parabola; and if e < 1, it defines an ellipse. The special case e = 0 of the latter results in a circle of radius
.
Differentiation
Differential calculus
We have the following formulas:
To find the Cartesian slope of the tangent line to a polar curve r(θ) at any given point, the curve is first expressed as a system of parametric equations.
Differentiating both equations with respect to θ yields
Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r, r(θ)):
Integration
Introduction
Integrating a polar equation requires a different approach than integration under the Cartesian system, hence yielding a different formula, which is not as straightforward as integrating the function .
Proof
In creating the concept of integration, we used Riemann sums of rectangles to approximate the area under the curve. However, with polar graphs, one can use sectors of circles with radius r and angle measure dθ. The area of each sector is then (πr²)(dθ/2π) and the sum of all the infinitesimally small sectors' areas is :, This is the form to use to integrate a polar expression of the form
where
and
are the ends of the curve that you wish to integrate.
Integral calculus
Let denote the region enclosed by a curve
and the rays
and
, where
. Then, the area of
is
This result can be found as follows. First, the interval is divided into
subintervals, where
is an arbitrary positive integer. Thus
, the length of each subinterval, is equal to
(the total length of the interval), divided by
, the number of subintervals. For each subinterval
, let
be the midpoint of the subinterval, and construct a circular sector with the center at the origin, radius
, central angle
, and arc length
. The area of each constructed sector is therefore equal to
. Hence, the total area of all of the sectors is
As the number of subintervals is increased, the approximation of the area continues to improve. In the limit as
, the sum becomes the Riemann integral.
Generalization
Using Cartesian coordinates, an infinitesimal area element can be calculated as =
. The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered:
Hence, an area element in polar coordinates can be written as
Now, a function that is given in polar coordinates can be integrated as follows:
Here, R is the same region as above, namely, the region enclosed by a curve and the rays
and
.
The formula for the area of mentioned above is retrieved by taking
identically equal to 1.
Applications
Polar integration is often useful when the corresponding integral is either difficult or impossible to do with the Cartesian coordinates. For example, let's try to find the area of the closed unit circle. That is, the area of the region enclosed by .
In Cartesian
In order to evaluate this, one usually uses trigonometric substitution. By setting , we get both
and
.
Putting this back into the equation, we get
In Polar
To integrate in polar coordinates, we first realize and in order to include the whole circle,
and
.
An interesting example
A less intuitive application of polar integration yields the Gaussian integral
Try it! (Hint: multiply and
.)
Sequences and Series
Sequences
A sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence.
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...).
Examples and notation
There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below.
A sequence may be denoted (a1, a2, ...). For shortness, the notation (an) is also used.
A more formal definition of a finite sequence with terms in a set S is a function from {1, 2, ..., n} to S for some n ≥ 0. An infinite sequence in S is a function from {1, 2, ...} (the set of natural numbers without 0) to S.
Sequences may also start from 0, so the first term in the sequence is then a0.
A finite sequence is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements.
A function from all integers into a set is sometimes called a bi-infinite sequence, since it may be thought of as a sequence indexed by negative integers grafted onto a sequence indexed by positive integers.
Types and properties of sequences
A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements (which, as stated in the introduction, can also be called "terms") without disturbing the relative positions of the remaining elements.
If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of a monotonic function. A sequence that both increases and decreases (at different places in the sequence) is said to be non-monotonic or non-monotone.
The terms non-decreasing and non-increasing are often used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. If the terms of a sequence are integers, then the sequence is an integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence.
If S is endowed with a topology (as is true of real numbers, for example), then it becomes possible to consider the convergence of an infinite sequence in S. Such considerations involve the concept of the limit of a sequence.
It can be shown that bounded monotonic sequences must converge.
Sequences in analysis
In analysis, when talking about sequences, one will generally consider sequences of the form
or
which is to say, infinite sequences of elements indexed by natural numbers. (It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N.)
The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers.
Series
Introduction
A series is the sum of a sequence of terms. An infinite series is the sum of an infinite number of terms (the actual sum of the series need not be infinite, as we will see below).
An arithmetic series is the sum of a sequence of terms with a common difference (the difference between consecutive terms). For example:
is an arithmetic series with common difference 3, since ,
, and so forth.
A geometric series is the sum of terms with a common ratio. For example, an interesting series which appears in many practical problems in science, engineering, and mathematics is the geometric series where the
indicates that the series continues indefinitely. A common way to study a particular series (following Cauchy) is to define a sequence consisting of the sum of the first
terms. For example, to study the geometric series we can consider the sequence which adds together the first n terms:
Generally by studying the sequence of partial sums we can understand the behavior of the entire infinite series.
Two of the most important questions about a series are:
- Does it converge?
- If so, what does it converge to?
For example, it is fairly easy to see that for , the geometric series
will not converge to a finite number (i.e., it will diverge to infinity). To see this, note that each time we increase the number of terms in the series,
increases by
, since
for all
(as we defined),
must increase by a number greater than one every term. When increasing the sum by more than one for every term, it will diverge.
Perhaps a more surprising and interesting fact is that for ,
will converge to a finite value. Specifically, it is possible to show that
Indeed, consider the quantity
Since as
for
, this shows that
as
. The quantity
is non-zero and doesn't depend on
so we can divide by it and arrive at the formula we want.
We'd like to be able to draw similar conclusions about any series.
Unfortunately, there is no simple way to sum a series. The most we will be able to do in most cases is determine if it converges. The geometric and the telescoping series are the only types of series in which we can easily find the sum of.
Convergence
It is obvious that for a series to converge, the an must tend to zero (because sum of an infinite number of terms all greater than any given positive number will be infinity), but even if the limit of the sequence is 0, this is not sufficient to say it converges.
Consider the harmonic series, the sum of 1/n, and group terms
As m tends to infinity, so does this final sum, hence the series diverges.
We can also deduce something about how quickly it diverges. Using the same grouping of terms, we can get an upper limit on the sum of the first so many terms, the partial sums.
or
and the partial sums increase like log m, very slowly.
Comparison test
The argument above, based on considering upper and lower bounds on terms, can be modified to provide a general-purpose test for convergence and divergence called the comparison test (or direct comparison test). It can be applied to any series with nonnegative terms:
- If
converges and
, then
converges.
- If
diverges and
, then
diverges.
There are many such tests for convergence and divergence, the most important of which we will describe below.
Absolute convergence
Theorem: If the series of absolute values, , converges, then so does the series
We say such a series converges absolutely.
Proof:
Let
According to the Cauchy criterion for series convergence, exists so that for all
:
We know that:
And then we get:
Now we get:
Which is exactly the Cauchy criterion for series convergence.
The converse does not hold. The series 1-1/2+1/3-1/4 ... converges, even though the series of its absolute values diverges.
A series like this that converges, but not absolutely, is said to converge conditionally.
If a series converges absolutely, we can add terms in any order we like. The limit will still be the same.
If a series converges conditionally, rearranging the terms changes the limit. In fact, we can make the series converge to any limit we like by choosing a suitable rearrangement.
E.g., in the series 1-1/2+1/3-1/4 ..., we can add only positive terms until the partial sum exceeds 100, subtract 1/2, add only positive terms until the partial sum exceeds 100, subtract 1/4, and so on, getting a sequence with the same terms that converges to 100.
This makes absolutely convergent series easier to work with. Thus, all but one of convergence tests in this chapter will be for series all of whose terms are positive, which must be absolutely convergent or divergent series. Other series will be studied by considering the corresponding series of absolute values.
Ratio test
For a series with terms an, if
then
- the series converges (absolutely) if r<1
- the series diverges if r>1 (or if r is infinity)
- the series could do either if r=1, so the test is not conclusive in this case.
E.g., suppose
then
so this series converges.
Integral test
If f(x) is a monotonically decreasing, always positive function, then the series
converges if and only if the integral
converges.
E.g., consider f(x)=1/xp, for a fixed p.
- If p=1 this is the harmonic series, which diverges.
- If p<1 each term is larger than the harmonic series, so it diverges.
- If p>1 then
The integral converges, for p>1, so the series converges.
We can prove this test works by writing the integral as
and comparing each of the integrals with rectangles, giving the inequalities
Applying these to the sum then shows convergence.
Limit comparison test
Given an infinite series with positive terms only, if one can find another infinite series
with positive terms for which
for a positive and finite L (i.e., the limit exists and is not zero), then the two series either both converge or both diverge. That is,
converges if
converges, and
diverges if
diverges.
Example:
For large n, the terms of this series are similar to, but smaller than, those of the harmonic series. We compare the limits.
so this series diverges.
Alternating series
Given an infinite series , if the signs of the an alternate, that is if
for all n or
for all n, then we call it an alternating series.
The alternating series test states that such a series converges if
and
(that is, the magnitude of the terms is decreasing).
Note that this test cannot lead to the conclusion that the series diverges; if one cannot conclude that the series converges, this test is inconclusive, although other tests may, of course, be used to give a conclusion.
Estimating the sum of an alternating series
The absolute error that results in using a partial sum of an alternating series to estimate the final sum of the infinite series is smaller than the magnitude of the first omitted term.
Geometric series
The geometric series can take either of the following forms
or
As you have seen at the start, the sum of the geometric series is
.
Telescoping series
Expanding (or "telescoping") this type of series is informative. If we expand this series, we get:
Additive cancellation leaves:
Thus,
and all that remains is to evaluate the limit.
There are other tests that can be used, but these tests are sufficient for all commonly encountered series.
Series and Calculus
Taylor Series
Taylor Series
The Taylor series of an infinitely, often differentiable real, (or complex) function f defined on an open interval (a-r, a+r) is the power series
Here, n! is the factorial of n and f (n)(a) denotes the nth derivative of f at the point a. If this series converges for every x in the interval (a-r, a+r) and the sum is equal to f(x), then the function f(x) is called analytic. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if a power series converges to the function; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.
If a = 0, the series is also called a Maclaurin series.
The importance of such a power series representation is threefold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. Third, the (truncated) series can be used to approximate values of the function near the point of expansion.
Note that there are examples of infinitely often differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, for the function defined piecewise by saying that f(x) = exp(−1/x²) if x ≠ 0 and f(0) = 0, all the derivatives are zero at x = 0, so the Taylor series of f(x) is zero, and its radius of convergence is infinite, even though the function most definitely is not zero. This particular pathology does not afflict complex-valued functions of a complex variable. Notice that exp(−1/z²) does not approach 0 as z approaches 0 along the imaginary axis.
Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = exp(−1/x²) can be written as a Laurent series.
The Parker-Sockacki theorem is a recent advance in finding Taylor series which are solutions to differential equations. This theorem is an expansion on the Picard iteration.
Derivation/why this works
If a function f(x) is written as a infinite power series, it will look like this:
f(x)=c0(x-a)0+c1(x-a)1+c2(x-a)2+c3(x-a)3+c4(x-a)4+c5(x-a)5+c6(x-a)6+c7(x-a)7+...
where a is half the radius of convergence and c0,c1,c2,c3,c4... are coefficients. If we substitute a for x:
f(a)=c0
If we differentiate:
f´(x)=1c1(x-a)0+2c2(x-a)1+3c3(x-a)2+4c4(x-a)3+5c5(x-a)4+6c6(x-a)5+7c7(x-a)6+...
If we substitute a for x:
f´(a)=1c1
If we differentiate:
f´´(x)=2c2+3*2*c3(x-a)1+4*3*c4(x-a)2+5*4*c5(x-a)3+6*5*c6(x-a)4+7*6*c7(x-a)5+...
If we substitute a for x:
f´´(a)=2c2
Extrapolating:
n!cn=fn(a)
where f0(x)=f(x) and f1(x)=f´(x) and so on.We can actually go ahead and say that the power approximation of f(x) is:
f(x)=Sn=0¥((fn(a)/n!)*(x-a)n)
<this needs to be improved>
List of Taylor series
Several important Taylor series expansions follow. All these expansions are also valid for complex arguments x.
Exponential function and natural logarithm:
The numbers Bk appearing in the expansions of tan(x) and tanh(x) are the Bernoulli numbers. The C(α,n) in the binomial expansion are the binomial coefficients. The Ek in the expansion of sec(x) are Euler numbers.
Multiple dimensions
The Taylor series may be generalized to functions of more than one variable with
History
The Taylor series is named for mathematician Brook Taylor, who first published the power series formula in 1715.
Constructing a Taylor Series
Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series (such as those above) to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. The use of computer algebra systems to calculate Taylor series is common, since it eliminates tedious substitution and manipulation.
Example 1
Consider the function
for which we want a Taylor series at 0.
We have for the natural logarithm
and for the cosine function
We can simply substitute the second series into the first. Doing so gives
Expanding by using multinomial coefficients gives the required Taylor series. Note that cosine and therefore are even functions, meaning that
, hence the coefficients of the odd powers
,
,
,
and so on have to be zero and don't need to be calculated. The first few terms of the series are
The general coefficient can be represented using Faà di Bruno's formula. However, this representation does not seem to be particularly illuminating and is therefore omitted here.
Example 2
Suppose we want the Taylor series at 0 of the function
We have for the exponential function
and, as in the first example,
Assume the power series is
Then multiplication with the denominator and substitution of the series of the cosine yields
Collecting the terms up to fourth order yields
Comparing coefficients with the above series of the exponential function yields the desired Taylor series
Convergence
Generalized Mean Value Theorem
Power Series
The study of power series is aimed at investigating series which can approximate some function over a certain interval.
Motivations
Elementary calculus (differentiation) is used to obtain information on a line which touches a curve at one point (i.e. a tangent). This is done by calculating the gradient, or slope of the curve, at a single point. However, this does not provide us with reliable information on the curve's actual value at given points in a wider interval. This is where the concept of power series becomes useful.
An example
Consider the curve of y = cos(x), about the point x = 0. A naïve approximation would be the line y = 1. However, for a more accurate approximation, observe that cos(x) looks like an inverted parabola around x = 0 - therefore, we might think about which parabola could approximate the shape of cos(x) near this point. This curve might well come to mind:
In fact, this is the best estimate for cos(x) which uses polynomials of degree 2 (i.e. a highest term of x2) - but how do we know this is true? This is the study of power series: finding optimal approximations to functions using polynomials.
Definition
A power series is a series of the form
- a0x0 + a1x1 + ... + anxn
or, equivalently,
Radius of convergence
When using a power series as an alternative method of calculating a function's value, the equation
can only be used to study f(x) where the power series converges - this may happen for a finite range, or for all real numbers.
The size of the interval (around its center) in which the power series converges to the function is known as the radius of convergence.
An example
(a geometric series)
this converges when | x | < 1, the range -1 < x < +1, so the radius of convergence - centered at 0 - is 1. It should also be observed that at the extremities of the radius, that is where x = 1 and x = -1, the power series does not converge.
Another example
Using the ratio test, this series converges when the ratio of successive terms is less than one:
- or
which is always true - therefore, this power series has an infinite radius of convergence. In effect, this means that the power series can always be used as a valid alternative to the original function, ex.
Abstraction
If we use the ratio test on an arbitrary power series, we find it converges when
and diverges when
The radius of convergence is therefore
If this limit diverges to infinity, the series has an infinite radius of convergence.
Differentiation and Integration
Within its radius of convergence, a power series can be differentiated and integrated term by term.
Both the differential and the integral have the same radius of convergence as the original series.
This allows us to sum exactly suitable power series. For example,
This is a geometric series, which converges for | x | < 1. Integrating both sides, we get
which will also converge for | x | < 1. When x = -1 this is the harmonic series, which diverges'; when x = 1 this is an alternating series with diminishing terms, which converges to ln 2 - this is testing the extremities.
It also lets us write power series for integrals we cannot do exactly such as the error function:
The left hand side can not be integrated exactly, but the right hand side can be.
This gives us a power series for the sum, which has an infinite radius of convergence, letting us approximate the integral as closely as we like.
Further reading
- "Decoding the Rosetta Stone" article by Jack W. Crenshaw 2005-10-12
Exercises
The following exercises test your understanding of infinite sequences and series. You may want to review that material before trying these problems.
Each question is followed by a "Hint" (usually a quick indication of the most efficient way to work the problem), the "Answer only" (what it sounds like), and finally a "Full solution" (showing all the steps required to get to the right answer). These should show up as "collapsed" or "hidden" sections (click on the title to display the contents), but some older web browsers might not be able to display them correctly (i.e., showing the content when it should be hidden). If this is true for your browser (or if you're looking at a printed version), you should take care not to "see too much" before you start thinking of how to work each problem.
Sequences
Consider the infinite sequence
- Is the sequence monotonically increasing or decreasing?
compare adjacent terms algebraically or take a derivative
monotonically decreasing (strictly decreasing starting with the second term)
One may either consider how compares to
algebraically and try to show that one is greater than the other, or take the derivative of
with respect to
and check where it is positive or negative.
Algebraically, since
we see that for
. That is, starting from the second term, the sequence is strictly decreasing. It is easy to check how the first two terms compare by just plugging in
and
:
The first two terms are equal and thereafter the terms are strictly decreasing. Therefore, the sequence is monotonically decreasing.
Using calculus,
which is negative for . The rest of the argument is the same as before.
- Is the sequence bounded from below, from above, both, or neither?
consider what kind of values are taken on by the numerator and denominator, and use the previous answer
bounded from below and from above (both)
The sequence is bounded from below because the terms are clearly positive (greater than 0) for all values of . Also, since the sequence is decreasing (see the previous problem), the maximum value of the sequence must be the value of the first term. So the sequence is bounded from above (by the value 1/2), as well.
- Does the sequence converge or diverge?
use the previous two answers to make a conclusion, or take a limit
converges
By the previous two answers, the sequence is bounded from below and monotonically decreasing, thus by a theorem it must converge.
To show this directly, consider the limit
The two limits are equal by L'Hôpital's Rule, since the numerator and denominator of the expression in the first limit both grow to infinity.
Since the limit exists, it is the number to which the sequence converges.
Partial sums
Assume that the nth partial sum of a series is given by
- Does the series converge? If so, to what value?
take a limit
converges to 2
The series converges to 2 since
- What is the formula for the nth term of the series?
Note that the series turns out to be geometric, since
Sums of infinite series
Find the value to which each of the following series converges.
sum of an infinite geometric series
4
The series is
and so is geometric with first term and common ratio
. So
sum of an infinite geometric series
telescoping series
1
rewrite so that all exponents are n
−1/5
The series simplifies to
and so is geometric with common ratio and first term
. Thus
Convergence and divergence of infinite series
Determine whether each the following series converges or diverges. (Note: Each "Hint" gives the convergence/divergence test required to draw a conclusion.)
p-series
converges
This is a p-series with . Since
, the series converges.
geometric series
converges
This is a geometric series with common ratio , and so converges since
.
limit comparison test
diverges
This series can be compared to a p-series:
The symbol means the two series are "asymptotically equivalent"—that is, they either both converge or both diverge because their terms behave so similarly when summed as n gets very large. This can be shown by the limit comparison test:
Since the limit is positive and finite, the two series either both converge or both diverge. The simpler series diverges because it is a p-series with (harmonic series), and so the original series diverges by the limit comparison test.
direct comparison test
diverges
This series can be compared to a smaller p-series:
The p-series diverges since (harmonic series), so the larger series diverges by the appropriate direct comparison test.
divergence test
diverges
The terms of this series do not have a limit of zero. Note that when ,
To see why the inequality holds, consider that when none of the fractions in the square brackets above are actually there; when
only 2/2 (which is the same as
) is in the brackets; when
only 3/2 (equal to
) and 2/2 (equal to
) are there; when
, only 4/2, 3/2, and 2/2 are there; and so forth. Clearly none of these fractions are less than 1 and they never will be, no matter what
is used.
The fact that
then implies that
Therefore the series diverges by the divergence test.
alternating series test
converges
This is an alternating series:
Since the sequence
decreases to 0, the series converges by the alternating series test.
alternating series test
converges
Since the terms alternate, consider the sequence
This sequence is clearly decreasing (since both n and are increasing — one may also show that the derivative (with respect to n) of the expression is negative for
) and has limit zero (the denominator goes to infinity), so the series converges by the alternating series test.
Absolute and conditional convergence
Determine whether each the following series converges conditionally, converges absolutely, or diverges. (Note: Each "Hint" gives the test or tests that most easily lead to the final conclusion.)
alternating series test and either direct comparison test or integral test
converges conditionally
This series alternates, so consider the sequence
Since this sequence is clearly decreasing to zero, the original series is convergent by the alternating series test. Now, consider the series formed by taking the absolute value of the terms of the original series:
This new series can be compared to a p-series:
Since the smaller series diverges, the larger one diverges. But this means the original (alternating) series was not absolutely convergent. (This last fact can also be shown using an integral test.) Therefore, the original series is only conditionally convergent.
alternating series test and either integral test or direct comparison test
converges conditionally
This series alternates, so consider the sequence
This sequence has a limit of zero by, for example, L'Hospital's Rule.
That the sequence is decreasing can be verified by, for example, showing that as a continuous function of x, its derivative is negative.
This means that the terms definitely decrease starting with the second term (). Thus, the series starting at
is convergent by the alternating series test; clearly, then, the series starting at
also converges (since the two series only differ by one term). Now, consider the series formed by taking the absolute value of the terms of the original series:
This new series of positive terms only can be compared to a p-series:
Since the smaller series diverges, the larger one diverges. Alternatively, the integral test can be used to test the convergence of the series of positive terms, since is clearly a continuous, positive function on
and, as we have just verified, is also decreasing:
by the substitution ; and this last expression becomes
Since the improper integral diverges, the series of positive terms diverges.
Either way you test it, the series with all positive terms diverges, and this means the original (alternating) series was not absolutely convergent. Thus, the original series is only conditionally convergent.
divergence test
diverges
This series is alternating, but note that by L'Hospital's Rule
Which implies that
does not exist, and hence by the divergence test, the series diverges.
limit comparison test with geometric series
converges absolutely
While this alternating series can be shown to converge by the alternating series test, it can also be shown that the absolute value of the terms form a convergent series, and this is sufficient to conclude absolute convergence of the original series. Thus we will skip the former test and show only the latter.
This series of positive terms is asymptotically geometric with :
The equivalence of these series is shown by using a limit comparison test:
Since the limit is positive and finite, and since the simpler series converges because it geometric with (the absolute value of which is less than 1), then the series of positive terms converges by the limit comparison test. Thus the original alternating series is absolutely convergent.
Note, by the way, that a direct comparison test in this case is more difficult (although still possible to do), since
and we need the inequality to go the other way to get a conclusion, since the geometric series converges.
This can be fixed by choosing the new series more carefully:
Comparing the original series to the new (convergent geometric) series gives the desired result.
divergence test
diverges
Since this is an alternating series, we can try the alternating series test. Consider the absolute value of the terms:
Because
does not exist (since it continually oscillates within the interval as n gets larger)
doesn't exist either. Thus the alternating series test fails (it is inconclusive).
However, in such a situation we can use the divergence test instead. Since
also does not exist (and thus the terms of the series do converge to 0), the original series diverges by the divergence test.
ratio test
converges absolutely
Because of the factorials in this series, we try the ratio test:
Since the limit is less than 1, the series converges absolutely by the ratio test.
divergence test
diverges
Although this is an alternating series, neither the numerator nor the denominator have infinite limits, so it is likely that a divergence test will work.
Note that
Thus
In fact, the latter limit does not exist. So, by the divergenve test, the series diverges.
Vector Calculations
Vectors
Two-Dimensional Vectors
Introduction
In most mathematics courses up until this point, we deal with scalars. These are quantities which only need one number to express. For instance, the amount of gasoline used to drive to the grocery store is a scalar quantity because it only needs one number: 2 gallons.
In this unit, we deal with vectors. A vector is a directed line segment -- that is, a line segment that points one direction or the other. As such, it has an initial point and a terminal point. The vector starts at the initial point and ends at the terminal point, and the vector points towards the terminal point. A vector is drawn as a line segment with an arrow at the terminal point:

The same vector can be placed anywhere on the coordinate plane and still be the same vector -- the only two bits of information a vector represents are the magnitude and the direction. The magnitude is simply the length of the vector, and the direction is the angle at which it points. Since neither of these specify a starting or ending location, the same vector can be placed anywhere. To illustrate, all of the line segments below can be defined as the vector with magnitude and angle 45 degrees:

It is customary, however, to place the vector with the initial point at the origin as indicated by the black vector. This is called the standard position.
Component Form
In standard practice, we don't express vectors by listing the length and the direction. We instead use component form, which lists the height (rise) and width (run) of the vectors. It is written as follows:

Other ways of denoting a vector in component form include:


From the diagram we can now see the benefits of the standard position: the two numbers for the terminal point's coordinates are the same numbers for the vector's rise and run. Note that we named this vector u. Just as you can assign numbers to variables in algebra (usually x, y, and z), you can assign vectors to variables in calculus. The letters u, v, and w are usually used, and either boldface or an arrow over the letter is used to identify it as a vector.
When expressing a vector in component form, it is no longer obvious what the magnitude and direction are. Therefore, we have to perform some calculations to find the magnitude and direction.
Magnitude

where is the width, or run, of the vector;
is the height, or rise, of the vector. You should recognize this formula as the Pythagorean theorem. It is -- the magnitude is the distance between the initial point and the terminal point.
The magnitude of a vector can also be called the norm.
Direction

where is the direction of the vector. This formula is simply the tangent formula for right triangles.
Vector Operations
For these definitions, assume:


Vector Addition
Vector Addition is often called tip-to-tail addition, because this makes it easier to remember.
The sum of the vectors you are adding is called the resultant vector, and is the vector drawn from the initial point (tip) of the first vector to the terminal point (tail) of the second vector. Although they look like the arrows, the pointy bit is the tail, not the tip. (Imagine you were walking the direction the vector was pointing... you would start at the flat end (tip) and walk toward the pointy end.)
It looks like this:
(Notice, the black lined vector is the sum of the two dotted line vectors!)
Numerically:
Or more generally:

Scalar Multiplication
Graphically, multiplying a vector by a scalar changes only the magnitude of the vector by that same scalar. That is, multiplying a vector by 2 will "stretch" the vector to twice its original magnitude, keeping the direction the same.


Numerically, you calculate the resultant vector with this formula:

As previously stated, the magnitude is changed by the same constant:

Since multiplying a vector by a constant results in a vector in the same direction, we can reason that two vectors are parallel if one is a constant multiple of the other -- that is, that if
for some constant c.
We can also divide by a non-zero scalar by instead multiplying by the reciprocal, as with dividing regular numbers:

Dot Product
The dot product is a way of multiplying two vectors to produce a scalar value. Because it combines the components of two vectors to form a /scalar/, it is sometimes called a scalar product. If you were asked to take the 'dot product of two rectangular vectors' you would do the following:

http://en.wikibooks.org/w/index.php?title=Calculus/Vectors&action=edit§ion=9
It is very important to note that the dot product of two vectors does not result in another vector, it gives you a scalar, just a numerical value.
Another common pitfall may arise if your vectors are not in rectangular ('cartesian') format. Sometimes, vectors are instead expressed in polar coordinates, where the first component is the vector's magnitude (length) and the second is the angle from the x-axis at which the vector should be oriented. Dot products cannot be performed using the conventional method on these sorts of vectors; vectors in polar format must be converted to their equivalent rectangular form before you can work with them using the formula given above. A common way to convert to rectangular coordinates is to imagine that the vector was projected horizontally and vertically to form a right triangle. You could then use properties of sin and cos to find the length of the two legs the right triangle. The horizontal length would then be the x-component of the rectangular expression of the vector and the vertical length would be the y-component. Remember that if the vector is pointing down or to the left, the corresponding components would have to be negative to indicate that.
With some rearrangement and trigonometric manipulation, we can see that the number that results from the dot product of two vectors is a surprising and useful identity:

where is the angle between the two vectors. This provides a convenient way of finding the angle between two vectors:

Notice that the dot product is 'commutative', that is:

Also, the dot product of two vectors will be the length of the vector squared:

and by the Pythagorean theorem,

The dot product can be visualized as the length of a projection of one vector on to the other. In other words, the dot product asks 'how much magnitude of this vector is going in the direction of that vector?'
Applications of Scalar Multiplication and Dot Product
Unit Vectors
A unit vector is a vector with a magnitude of 1. The unit vector of u is a vector in the same direction as , but with a magnitude of 1:

The process of finding the unit vector of u is called normalization. As mentioned in scalar multiplication, multiplying a vector by constant C will result in the magnitude being multiplied by C. We know how to calculate the magnitude of . We know that dividing a vector by a constant will divide the magnitude by that constant. Therefore, if that constant is the magnitude, dividing the vector by the magnitude will result in a unit vector in the same direction as
:
, where
is the unit vector of
Standard Unit Vectors
A special case of Unit Vectors are the Standard Unit Vectors i and j: i points one unit directly right in the x direction, and j points one unit directly up in the y direction:
Using the scalar multiplication and vector addition rules, we can then express vectors in a different way:
If we work that equation out, it makes sense. Multiplying x by i will result in the vector . Multiplying y by j will result in the vector
. Adding these two together will give us our original vector,
. Expressing vectors using i and j is called standard form.
Projection and Decomposition of Vectors
Sometimes it is necessary to decompose a vector into two components: one component parallel to a vector
, which we will call
; and one component perpendicular to it,
.

Since the length of is (
), it is straightforward to write down the formulas for
and
:

and

Length of a vector
The length of a vector is given by the dot product of a vector with itself, and :

Perpendicular vectors
If the angle between two vectors is 90 degrees or
(if the two vectors are orthogonal to each other), that is the vectors are perpendicular, then the dot product is 0. This provides us with an easy way to find a perpendicular vector: if you have a vector
, a perpendicular vector can easily be found by either

Polar coordinates
Polar coordinates are an alternative two-dimensional coordinate system, which is often useful when rotations are important. Instead of specifying the position along the x and y axes, we specify the distance from the origin, r, and the direction, an angle θ.
Looking at this diagram, we can see that the values of x and y are related to those of r and θ by the equations

Because tan-1 is multivalued, care must be taken to select the right value.
Just as for Cartesian coordinates the unit vectors that point in the x and y directions are special, so in polar coordinates the unit vectors that point in the r and θ directions are also special.
We will call these vectors and
, pronounced r-hat and theta-hat. Putting a circumflex over a vector this way is often used to mean the unit vector in that direction.
Again, on looking at the diagram we see,
Three-Dimensional Coordinates and Vectors
Basic definition
Two-dimensional Cartesian coordinates as we've discussed so far can be easily extended to three-dimensions by adding one more value: 'z'. If the standard (x,y) coordinate axes are drawn on a sheet of paper, the 'z' axis would extend upwards off of the paper.
Similar to the two coordinate axes in two-dimensional coordinates, there are three coordinate planes in space. These are the xy-plane, the yz-plane, and the xz-plane. Each plane is the "sheet of paper" that contains both axes the name mentions. For instance, the yz-plane contains both the y and z axes and is perpendicular to the x axis.
center|Coordinate planes in space.
Therefore, vectors can be extended to three dimensions by simply adding the 'z' value.

To facilitate standard form notation, we add another standard unit vector:

Again, both forms (component and standard) are equivalent.

Magnitude: Magnitude in three dimensions is the same as in two dimensions, with the addition of a 'z' term in the radicand.

Three dimensions
The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate systems, both of which include two-dimensional or planar polar coordinates as a subset. In essence, the cylindrical coordinate system extends polar coordinates by adding an additional distance coordinate, while the spherical system instead adds an additional angular coordinate.
Cylindrical coordinates
The cylindrical coordinate system is a coordinate system that essentially extends the two-dimensional polar coordinate system by adding a third coordinate measuring the height of a point above the plane, similar to the way in which the Cartesian coordinate system is extended into three dimensions. The third coordinate is usually denoted h, making the three cylindrical coordinates (r, θ, h).
The three cylindrical coordinates can be converted to Cartesian coordinates by
Spherical coordinates
Polar coordinates can also be extended into three dimensions using the coordinates (ρ, φ, θ), where ρ is the distance from the origin, φ is the angle from the z-axis (called the colatitude or zenith and measured from 0 to 180°) and θ is the angle from the x-axis (as in the polar coordinates). This coordinate system, called the spherical coordinate system, is similar to the latitude and longitude system used for Earth, with the origin in the centre of Earth, the latitude δ being the complement of φ, determined by δ = 90° − φ, and the longitude l being measured by l = θ − 180°.
The three spherical coordinates are converted to Cartesian coordinates by
Cross Product
The cross product of two vectors is a determinant:
and is also a pseudovector.
The cross product of two vectors is orthogonal to both vectors. The magnitude of the cross product is the product of the magnitude of the vectors and the sin of the angle between them.
This magnitude is the area of the parallelogram defined by the two vectors.
The cross product is linear and anticommutative. For any numbers a and b,
If both vectors point in the same direction, their cross product is zero.
Triple Products
If we have three vectors we can combine them in two ways, a triple scalar product,

and a triple vector product

The triple scalar product is a determinant
If the three vectors are listed clockwise, looking from the origin, the sign of this product is positive. If they are listed anticlockwise the sign is negative.
The order of the cross and dot products doesn't matter.
Either way, the absolute value of this product is the volume of the parallelepiped defined by the three vectors, u, v, and w
The triple vector product can be simplified

This form is easier to do calculations with.
The triple vector product is not associative.

There are special cases where the two sides are equal, but in general the brackets matter. They must not be omitted.
Three-Dimensional Lines and Planes
We will use r to denote the position of a point.
The multiples of a vector, a all lie on a line through the origin. Adding a constant vector b will shift the line, but leave it straight, so the equation of a line is,
This is a parametric equation. The position is specified in terms of the parameter s.
Any linear combination of two vectors, a and b lies on a single plane through the origin, provided the two vectors are not colinear. We can shift this plane by a constant vector again and write
If we choose a and b to be orthonormal vectors in the plane (i.e. unit vectors at right angles) then s and t are Cartesian coordinates for points in the plane.
These parametric equations can be extended to higher dimensions.
Instead of giving parametric equations for the line and plane, we could use constraints. E.g., for any point in the xy plane z=0
For a plane through the origin, the single vector normal to the plane, n, is at right angle with every vector in the plane, by definition, so
is a plane through the origin, normal to n.
For planes not through the origin we get
A line lies on the intersection of two planes, so it must obey the constraint for both planes, i.e.
These constraint equations con also be extended to higher dimensions.
Vector-Valued Functions
Vector-Valued Functions are functions that instead of giving a resultant scalar value, give a resultant vector value. These aid in the creation of direction and vector fields, and are therefore used in physics to aid with visualizations of electric, magnetic, and many other types of fields. They are of the following form:
Introduction
Limits, Derivatives, and Integrals
Put simply, the limit of a vector-valued function is the limit of its parts.
Proof:
Suppose
Therefore for any there is a
such that
But by the triangle inequality
So
Therefore A similar argument can be used through parts a_n(t)
Now let again, and that for any ε>0 there is a corresponding φ>0 such 0<|t-c|<φ implies
Then
therefore!:
From this we can then create an accurate definition of a derivative of a vector-valued function:
The final step was accomplished by taking what we just did with limits.
By the Fundamental Theorem of Calculus integrals can be applied to the vector's components.
In other words: the limit of a vector function is the limit of its parts, the derivative of a vector function is the derivative of its parts, and the integration of a vector function is the integration of it parts.
Velocity, Acceleration, Curvature, and a brief mention of the Binormal
Assume we have a vector-valued function which starts at the origin and as its independent variables changes the points that the vectors point at trace a path.
We will call this vector , which is commonly known as the position vector.
If then represents a position and t represents time, then in model with Physics we know the following:
is displacement.
where
is the velocity vector.
is the speed.
where
is the acceleration vector.
The only other vector that comes in use at times is known as the curvature vector.
The vector used to find it is known as the unit tangent vector, which is defined as
or shorthand
.
The vector normal to this then is
.
We can verify this by taking the dot product
Also note that
and
and
Then we can actually verify:
Therefore is perpendicular to
What this gives rise to is the Unit Normal Vector of which the top-most vector is the Normal vector, but the bottom half
is known as the curvature. Since the Normal vector points toward the inside of a curve, the sharper a turn, the Normal vector has a large magnitude, therefore the curvature has a small value, and is used as an index in civil engineering to reflect the sharpness of a curve (clover-leaf highways, for instance).
The only other thing not mentioned is the Binormal that occurs in 3-d curves , which is useful in creating planes parallel to the curve.
Lines and Planes in Space
Introduction
For many practical applications, for example for describing forces in physics and mechanics, you have to work with the mathematical descriptions of lines and planes in 3-dimensional space.
Parametric Equations
Line in Space
A line in space is defined by two points in space, which I will call and
. Let
be the vector from the origin to
, and
the vector from the origin to
. Given these two points, every other point
on the line can be reached by

where is the vector from
and
:

Plane in Space
The same idea can be used to describe a plane in 3-dimensional space, which is uniquely defined by three points (which do not lie on a line) in space (). Let
be the vectors from the origin to
. Then

with:

Note that the starting point does not have to be , but can be any point in the plane. Similarly, the only requirement on the vectors
and
is that they have to be two non-collinear vectors in our plane.
Vector Equation (of a Plane in Space, or of a Line in a Plane)
An alternative representation of a Plane in Space is obtained by observing that a plane is defined by a point in that plane and a direction perpendicular to the plane, which we denote with the vector
. As above, let
describe the vector from the origin to
, and
the vector from the origin to another point
in the plane. Since any vector that lies in the plane is perpendicular to
, the vector equation of the plane is given by

In 2 dimensions, the same equation uniquely describes a Line.
Scalar Equation (of a Plane in Space, or of a Line in a Plane)
If we express and
through their components

writing out the scalar product for provides us with the scalar equation for a plane in space:

where .
In 2d space, the equivalent steps lead to the scalar equation for a line in a plane:

Multivariable & Differential Calculus
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In your previous study of calculus, we have looked at functions and their behavior. Most of these functions we have examined have been all in the form
- f(x) : R → R,
and only occasional examination of functions of two variables. However, the study of functions of several variables is quite rich in itself, and has applications in several fields.
We write functions of vectors - many variables - as follows:
- f : Rm → Rn
and f(x) for the function that maps a vector in Rm to a vector in Rn.
Before we can do calculus in Rn, we must familiarize ourselves with the structure of Rn. We need to know which properties of R can be extended to Rn
Topology in Rn
We are already familiar with the nature of the regular real number line, which is the set R, and the two-dimensional plane, R2. This examination of topology in Rn attempts to look at a generalization of the nature of n-dimensional spaces - R, or R23, or Rn.
Lengths and distances
If we have a vector in R2, we can calculate its length using the Pythagorean theorem. For instance, the length of the vector (2, 3) is
We can generalize this to Rn. We define a vector's length, written |x|, as the square root of the sum of the squares of each of its components. That is, if we have a vector x=(x1,...,xn),
Now that we have established some concept of length, we can establish the distance between two vectors. We define this distance to be the length of the two vectors' difference. We write this distance d(x, y), and it is
This distance function is sometimes referred to as a metric. Other metrics arise in different circumstances. The metric we have just defined is known as the Euclidean metric.
Open and closed balls
In R, we have the concept of an interval, in that we choose a certain number of other points about some central point. For example, the interval [-1, 1] is centered about the point 0, and includes points to the left and right of zero.
In R2 and up, the idea is a little more difficult to carry on. For R2, we need to consider points to the left, right, above, and below a certain point. This may be fine, but for R3 we need to include points in more directions.
We generalize the idea of the interval by considering all the points that are a given, fixed distance from a certain point - now we know how to calculate distances in Rn, we can make our generalization as follows, by introducing the concept of an open ball and a closed ball respectively, which are analogous to the open and closed interval respectively.
- an open ball
- is a set in the form { x ∈ Rn|d(x, a) < r}
- a closed ball
- is a set in the form { x ∈ Rn|d(x, a) ≤ r}
In R, we have seen that the open ball is simply an open interval centered about the point x=a. In R2 this is a circle with no boundary, and in R3 it is a sphere with no outer surface. (What would the closed ball be?)
Boundary points
If we have some area, say a field, then the common sense notion of the boundary is the points 'next to' both the inside and outside of the field. For a set, S, we can define this rigorously by saying the boundary of the set contains all those points such that we can find points both inside and outside the set. We call the set of such points ∂S
Typically, when it exists the dimension of ∂S is one lower than the dimension of S. e.g. the boundary of a volume is a surface and the boundary of a surface is a curve.
This isn't always true; but it is true of all the sets we will be using.
A set S is bounded if there is some positive number such that we can encompass this set by a closed ball about 0. --> if every point in it is within a finite distance of the origin, i.e there exists some r>0 such that x is in S implies |x|<r.
Curves and parameterizations
If we have a function f : R → Rn, we say that f's image (the set {f(t) | t ∈ R} - or some subset of R) is a curve in Rn and f is its parametrization.
Parameterizations are not necessarily unique - for example, f(t) = (cos t, sin t) such that t ∈ [0, 2π) is one parametrization of the unit circle, and g(t) = (cos at, sin at) such that t ∈ [0, 2π/a) is a whole family of parameterizations of that circle.
Collision and intersection points
Say we have two different curves. It may be important to consider
- points the two curves share - where they intersect
- intersections which occur for the same value of t - where they collide.
Intersection points
Firstly, we have two parameterizations f(t) and g(t), and we want to find out when they intersect, this means that we want to know when the function values of each parametrization are the same. This means that we need to solve
- f(t) = g(s)
because we're seeking the function values independent of the times they intersect.
For example, if we have f(t) = (t, 3t) and g(t) = (t, t2), and we want to find intersection points:
- f(t) = g(s)
- (t, 3t) = (s, s2)
- t = s and 3t = s2
with solutions (t, s) = (0, 0) and (3, 3)
So, the two curves intersect at the points (0, 0) and (3, 3).
Collision points
However, if we want to know when the points "collide", with f(t) and g(t), we need to know when both the function values and the times are the same, so we need to solve instead
- f(t) = g(t)
For example, using the same functions as before, f(t) = (t, 3t) and g(t) = (t, t2), and we want to find collision points:
- f(t) = g(t)
- (t, 3t) = (t, t2)
- t = t and 3t = t2
which gives solutions t = 0, 3 So the collision points are (0, 0) and (3, 3).
We may want to do this to actually model physical problems, such as in ballistics.
Continuity and differentiability
If we have a parametrization f : R → Rn, which is built up out of component functions in the form f(t) = (f1(t),...,fn(t)), f is continuous if and only if each component function is also.
In this case the derivative of f(t) is
- ai = (f1′(t),...,fn′(t)). This is actually a specific consequence of a more general fact we will see later.
Tangent vectors
Recall in single-variable calculus that on a curve, at a certain point, we can draw a line that is tangent to that curve at exactly at that point. This line is called a tangent. In the several variable case, we can do something similar.
We can expect the tangent vector to depend on f′(t) and we know that a line is its own tangent, so looking at a parametrised line will show us precisely how to define the tangent vector for a curve.
An arbitrary line is f(t)=at+b, with :fi(t)=ait+bi, so
- fi′(t)=ai and
- f′(t)=a, which is the direction of the line, its tangent vector.
Similarly, for any curve, the tangent vector is f′(t).
Angle between curves
We can then formulate the concept of the angle between two curves by considering the angle between the two tangent vectors. If two curves, parametrized by f1 and f2 intersect at some point, which means that
- f1(s)=f2(t)=c,
the angle between these two curves at c is the angle between the tangent vectors f1′(s) and f2′(t) is given by
Tangent lines
With the concept of the tangent vector as being analogous to being the gradient of the line in the one variable case, we can form the idea of the tangent line. Recall that we need a point on the line and its direction.
If we want to form the tangent line to a point on the curve, say p, we have the direction of the line f′(p), so we can form the tangent line
- x(t)=p+t f′(p)
Different parameterizations
One such parametrization of a curve is not necessarily unique. Curves can have several different parametrizations. For example, we already saw that the unit circle can be parametrized by g(t) = (cos at, sin at) such that t ∈ [0, 2π/a).
Generally, if f is one parametrization of a curve, and g is another, with
- f(t0) = g(s0)
there is a function u(t) such that u(t0)=s0, and g'(u(t)) = f(t) near t0.
This means, in a sense, the function u(t) "speeds up" the curve, but keeps the curve's shape.
Surfaces
A surface in space can be described by the image of a function f : R2 → Rn. f is said to be the parametrization of that surface.
For example, consider the function
- f(α, β) = α(2,1,3)+β(-1,2,0)
This describes an infinite plane in R3. If we restrict α and β to some domain, we get a parallelogram-shaped surface in R3.
Surfaces can also be described explicitly, as the graph of a function z = f(x, y) which has a standard parametrization as f(x,y)=(x, y, f(x,y)), or implictly, in the form f(x, y, z)=c.
Level sets
The concept of the level set (or contour) is an important one. If you have a function f(x, y, z), a level set in R3 is a set of the form {(x,y,z)|f(x,y,z)=c}. Each of these level sets is a surface.
Level sets can be similarly defined in any Rn
Level sets in two dimensions may be familiar from maps, or weather charts. Each line represents a level set. For example, on a map, each contour represents all the points where the height is the same. On a weather chart, the contours represent all the points where the air pressure is the same.
Limits and continuity
Before we can look at derivatives of multivariate functions, we need to look at how limits work with functions of several variables first, just like in the single variable case.
If we have a function f : Rm → Rn, we say that f(x) approaches b (in Rn) as x approaches a (in Rm) if, for all positive ε, there is a corresponding positive number δ, |f(x)-b| < ε whenever |x-a| < δ, with x ≠ a.
This means that by making the difference between x and a smaller, we can make the difference between f(x) and b as small as we want.
If the above is true, we say
- f(x) has limit b at a
- f(x) approaches b as x approaches a
- f(x) → b as x → a
These four statements are all equivalent.
Rules
Since this is an almost identical formulation of limits in the single variable case, many of the limit rules in the one variable case are the same as in the multivariate case.
For f and g, mapping Rm to Rn, and h(x) a scalar function mapping Rm to R, with
- f(x) → b as x → a
- g(x) → c as x → a
- h(x) → H as x → a
then:
and consequently
when H≠0
Continuity
Again, we can use a similar definition to the one variable case to formulate a definition of continuity for multiple variables.
If f : Rm → Rn, f is continuous at a point a in Rm if f(a) is defined and
Just as for functions of one dimension, if f, g are both continuous at p, f+g, λf (for a scalar λ), f·g, and f×g are continuous also. If φ : Rm → R is continus at p, φf, f/φ are too if φ is never zero.
From these facts we also have that if A is some matrix which is n×m in size, with x in Rm, a function f(x)=A x is continuous in that the function can be expanded in the form x1a1+...+xmam, which can be easily verified from the points above.
If f : Rm → Rn which is in the form f(x) = (f1(x),...,fn(x) is continuous if and only if each of its component functions are a polynomial or rational function, whenever they are defined.
Finally, if f is continuous at p, g is continuous at f(p), g(f(x)) is continuous at p.
Special note about limits
It is important to note that we can approach a point in more than one direction, and thus, the direction that we approach that point counts in our evaluation of the limit. It may be the case that a limit may exist moving in one direction, but not in another.
Differentiable functions
We will start from the one-variable definition of the derivative at a point p, namely
Let's change above to equivalent form of
which achieved after pulling f'(p) inside and putting it over a common denominator.
We can't divide by vectors, so this definition can't be immediately extended to the multiple variable case. Nonetheless, we don't have to: the thing we took interest in was the quotient of two small distances (magnitudes), not their other properties (like sign). It's worth noting that 'other' property of vector neglected is its direction. Now we can divide by the absolute value of a vector, so lets rewrite this definition in terms of absolute values
Another form of formula above is obtained by letting we have
and if
, the
, so
,
where can be thought of as a 'small change'.
So, how can we use this for the several-variable case?
If we switch all the variables over to vectors and replace the constant (which performs a linear map in one dimension) with a matrix (which denotes also a linear map), we have
or
If this limit exists for some f : Rm → Rn, and there is a linear map A : Rm → Rn (denoted by matrix A which is m×n), we refer to this map as being the derivative and we write it as Dp f.
A point on terminology - in referring to the action of taking the derivative (giving the linear map A), we write Dp f, but in referring to the matrix A itself, it is known as the Jacobian matrix and is also written Jp f. More on the Jacobian later.
Properties
There are a number of important properties of this formulation of the derivative.
Affine approximations
If f is differentiable at p for x close to p, |f(x)-(f(p)+A(x-p))| is small compared to |x-p|, which means that f(x) is approximately equal to f(p)+A(x-p).
We call an expression of the form g(x)+c affine, when g(x) is linear and c is a constant. f(p)+A(x-p) is an affine approximation to f(x).
Jacobian matrix and partial derivatives
The Jacobian matrix of a function is in the form
for a f : Rm → Rn, Jp f' is a n×m matrix.
The consequence of this is that if f is differentiable at p, all the partial derivatives of f exist at p.
However, it is possible that all the partial derivatives of a function exist at some point yet that function is not differentiable there, so it is very important not to mix derivative (linear map) with the Jacobian (matrix) especially in situations akin to the one cited.
Continuity and differentiability
Furthermore, if all the partial derivatives exist, and are continuous in some neighbourhood of a point p, then f is differentiable at p. This has the consequence that for a function f which has its component functions built from continuous functions (such as rational functions, differentiable functions or otherwise), f is differentiable everywhere f is defined.
We use the terminology continuously differentiable for a function differentiable at p which has all its partial derivatives existing and are continuous in some neighbourhood at p.
Rules of taking Jacobians
If f : Rm → Rn, and h(x) : Rm → R are differentiable at 'p':
Important: make sure the order is right - matrix multiplication is not commutative!
Chain rule
The chain rule for functions of several variables is as follows. For f : Rm → Rn and g : Rn → Rp, and g o f differentiable at p, then the Jacobian is given by
Again, we have matrix multiplication, so one must preserve this exact order. Compositions in one order may be defined, but not necessarily in the other way.
Alternate notations
For simplicity, we will often use various standard abbreviations, so we can write most of the formulae on one line. This can make it easier to see the important details.
We can abbreviate partial differentials with a subscript, e.g.,
When we are using a subscript this way we will generally use the Heaviside D rather than ∂,
Mostly, to make the formulae even more compact, we will put the subscript on the function itself.
If we are using subscripts to label the axes, x1, x2 …, then, rather than having two layers of subscripts, we will use the number as the subscript.
We can also use subscripts for the components of a vector function, u=(ux, uy, uy) or u=(u1,u2…un)
If we are using subscripts for both the components of a vector and for partial derivatives we will separate them with a comma.
The most widely used notation is hx. Both h1 and ∂1h are also quite widely used whenever the axes are numbered. The notation ∂xh is used least frequently.
We will use whichever notation best suits the equation we are working with.
Directional derivatives
Normally, a partial derivative of a function with respect to one of its variables, say, xj, takes the derivative of that "slice" of that function parallel to the xj'th axis.
More precisely, we can think of cutting a function f(x1,...,xn) in space along the xj'th axis, with keeping everything but the xj variable constant.
From the definition, we have the partial derivative at a point p of the function along this slice as
provided this limit exists.
Instead of the basis vector, which corresponds to taking the derivative along that axis, we can pick a vector in any direction (which we usually take as being a unit vector), and we take the directional derivative of a function as
where d is the direction vector.
If we want to calculate directional derivatives, calculating them from the limit definition is rather painful, but, we have the following: if f : Rn → R is differentiable at a point p, |p|=1,
There is a closely related formulation which we'll look at in the next section.
Gradient vectors
The partial derivatives of a scalar tell us how much it changes if we move along one of the axes. What if we move in a different direction?
We'll call the scalar f, and consider what happens if we move an infintesimal direction dr=(dx,dy,dz), using the chain rule.
This is the dot product of dr with a vector whose components are the partial derivatives of f, called the gradient of f
We can form directional derivatives at a point p, in the direction d then by taking the dot product of the gradient with d
.
Notice that grad f looks like a vector multiplied by a scalar. This particular combination of partial derivatives is commonplace, so we abbreviate it to
We can write the action of taking the gradient vector by writing this as an operator. Recall that in the one-variable case we can write d/dx for the action of taking the derivative with respect to x. This case is similar, but ∇ acts like a vector.
We can also write the action of taking the gradient vector as:
Properties of the gradient vector
Geometry
- Grad f(p) is a vector pointing in the direction of steepest slope of f. |grad f(p)| is the rate of change of that slope at that point.
For example, if we consider h(x, y)=x2+y2. The level sets of h are concentric circles, centred on the origin, and
grad h points directly away from the origin, at right angles to the contours.
- Along a level set, (∇f)(p) is perpendicular to the level set {x|f(x)=f(p) at x=p}.
If dr points along the contours of f, where the function is constant, then df will be zero. Since df is a dot product, that means that the two vectors, df and grad f, must be at right angles, i.e. the gradient is at right angles to the contours.
Algebraic properties
Like d/dx, ∇ is linear. For any pair of constants, a and b, and any pair of scalar functions, f and g
Since it's a vector, we can try taking its dot and cross product with other vectors, and with itself.
Divergence
If the vector function u maps Rn to itself, then we can take the dot product of u and ∇. This dot product is called the divergence.
If we look at a vector function like v=(1+x2,xy) we can see that to the left of the origin all the v vectors are converging towards the origin, but on the right they are diverging away from it.
Div u tells us how much u is converging or diverging. It is positive when the vector is diverging from some point, and negative when the vector is converging on that point.
- Example:
- For v=(1+x2, xy), div v=3x, which is positive to the right of the origin, where v is diverging, and negative to the left of the origin, where v is converging.
Like grad, div is linear.
Later in this chapter we will see how the divergence of a vector function can be integrated to tell us more about the behaviour of that function.
To find the divergence we took the dot product of ∇ and a vector with ∇ on the left. If we reverse the order we get
To see what this means consider i·∇ This is Dx, the partial differential in the i direction. Similarly, u·∇ is the partial differential in the u direction, multiplied by |u|
Curl
If u is a three-dimensional vector function on R3 then we can take its cross product with ∇. This cross product is called the curl.
Curl u tells us if the vector u is rotating round a point. The direction of curl u is the axis of rotation.
We can treat vectors in two dimensions as a special case of three dimensions, with uz=0 and Dzu=0. We can then extend the definition of curl u to two-dimensional vectors
This two dimensional curl is a scalar. In four, or more, dimensions there is no vector equivalent to the curl.
Example:
Consider u=(-y, x). These vectors are tangent to circles centred on the origin, so appear to be rotating around it anticlockwise.
Example
Consider u=(-y, x-z, y), which is similar to the previous example.
This u is rotating round the axis i+k
Later in this chapter we will see how the curl of a vector function can be integrated to tell us more about the behaviour of that function.
Product and chain rules
Just as with ordinary differentiation, there are product rules for grad, div and curl.
- If g is a scalar and v is a vector, then
-
- the divergence of gv is
- the curl of gv is
- If u and v are both vectors then
-
- the gradient of their dot product is
- the divergence of their cross product is
- the curl of their cross product is
We can also write chain rules. In the general case, when both functions are vectors and the composition is defined, we can use the Jacobian defined earlier.
where Ju is the Jacobian of u at the point v.
Normally J is a matrix but if either the range or the domain of u is R1 then it becomes a vector. In these special cases we can compactly write the chain rule using only vector notation.
- If g is a scalar function of a vector and h is a scalar function of g then
- If g is a scalar function of a vector then
This substitution can be made in any of the equations containing ∇
Second order differentials
We can also consider dot and cross products of ∇ with itself, whenever they can be defined. Once we know how to simplify products of two ∇'s we'll know out to simplify products with three or more.
The divergence of the gradient of a scalar f is
This combination of derivatives is the Laplacian of f. It is commmonplace in physics and multidimensional calculus because of its simplicity and symmetry.
We can also take the Laplacian of a vector,
The Laplacian of a vector is not the same as the divergence of its gradient
Both the curl of the gradient and the divergence of the curl are always zero.
This pair of rules will prove useful.
Integration
We have already considered differentiation of functions of more than one variable, which leads us to consider how we can meaningfully look at integration.
In the single variable case, we interpret the definite integral of a function to mean the area under the function. There is a similar interpretation in the multiple variable case: for example, if we have a paraboloid in R3, we may want to look at the integral of that paraboloid over some region of the xy plane, which will be the volume under that curve and inside that region.
Riemann sums
When looking at these forms of integrals, we look at the Riemann sum. Recall in the one-variable case we divide the interval we are integrating over into rectangles and summing the areas of these rectangles as their widths get smaller and smaller. For the multiple-variable case, we need to do something similar, but the problem arises how to split up R2, or R3, for instance.
To do this, we extend the concept of the interval, and consider what we call a n-interval. An n-interval is a set of points in some rectangular region with sides of some fixed width in each dimension, that is, a set in the form {x∈Rn|ai ≤ xi ≤ bi with i = 0,...,n}, and its area/size/volume (which we simply call its measure to avoid confusion) is the product of the lengths of all its sides.
So, an n-interval in R2 could be some rectangular partition of the plane, such as {(x,y) | x ∈ [0,1] and y ∈ [0, 2]|}. Its measure is 2.
If we are to consider the Riemann sum now in terms of sub-n-intervals of a region Ω, it is
where m(Si) is the measure of the division of Ω into k sub-n-intervals Si, and x*i is a point in Si. The index is important - we only perform the sum where Si falls completely within Ω - any Si that is not completely contained in Ω we ignore.
As we take the limit as k goes to infinity, that is, we divide up Ω into finer and finer sub-n-intervals, and this sum is the same no matter how we divide up Ω, we get the integral of f over Ω which we write
For two dimensions, we may write
and likewise for n dimensions.
Iterated integrals
Thankfully, we need not always work with Riemann sums every time we want to calculate an integral in more than one variable. There are some results that make life a bit easier for us.
For R2, if we have some region bounded between two functions of the other variable (so two functions in the form f(x) = y, or f(y) = x), between a constant boundary (so, between x = a and x =b or y = a and y = b), we have
An important theorem (called Fubini's theorem) assures us that this integral is the same as
,
if f is continuous on the domain of integration.
Order of integration
In some cases the first integral of the entire iterated integral is difficult or impossible to solve, therefore, it can be to our advantage to change the order of integration.
As of the writing of this, there is no set method to change an order of integration from dxdy to dydx or some other variable. Although, it is possible to change the order of integration in an x and y simple integration by simply switching the limits of integration around also, in non-simple x and y integrations the best method as of yet is to recreate the limits of the integration from the graph of the limits of integration.
In higher order integration that can't be graphed, the process can be very tedious. For example, dxdydz can be written into dzdydx, but first dxdydz must be switched to dydxdz and then to dydzdx and then to dzdydx (as 3-dimensional cases can be graphed, this method would lack parsimony).
Parametric integrals
If we have a vector function, u, of a scalar parameter, s, we can integrate with respect to s simply by integrating each component of u separately.
Similarly, if u is given a function of vector of parameters, s, lying in Rn, integration with respect to the parameters reduces to a multiple integral of each component.
Line integrals
In one dimension, saying we are integrating from a to b uniquely specifies the integral.
In higher dimensions, saying we are integrating from a to b is not sufficient. In general, we must also specify the path taken between a and b.
We can then write the integrand as a function of the arclength along the curve, and integrate by components.
E.g., given a scalar function h(r) we write
where C is the curve being integrated along, and t is the unit vector tangent to the curve.
There are some particularly natural ways to integrate a vector function, u, along a curve,
where the third possibility only applies in 3 dimensions.
Again, these integrals can all be written as integrals with respect to the arclength, s.
If the curve is planar and u a vector lying in the same plane, the second integral can be usefully rewritten. Say,
where t, n, and b are the tangent, normal, and binormal vectors uniquely defined by the curve.
Then
For the 2-d curves specified b is the constant unit vector normal to their plane, and ub is always zero.
Therefore, for such curves,
Green's Theorem
Let C be a piecewise smooth, simple closed curve that bounds a region S on the Cartesian plane. If two function M(x,y) and N(x,y) are continuous and their partial derivatives are continuous, then
In order for Green's theorem to work there must be no singularities in the vector field within the boundaries of the curve.
Green's theorem works by summing the circulation in each infinitesimal segment of area enclosed within the curve.
Inverting differentials
We can use line integrals to calculate functions with specified divergence, gradient, or curl.
- If grad V = u
- where h is any function of zero gradient and curl u must be zero.
- If div u = V
- where w is any function of zero divergence.
- If curl u = v
- where w is any function of zero curl.
For example, if V=r2 then
and
so this line integral of the gradient gives the original function.
Similarly, if v=k then
Consider any curve from 0 to p=(x, y', z), given by r=r(s) with r(0)=0 and r(S)=p for some S, and do the above integral along that curve.
and curl u is
as expected.
We will soon see that these three integrals do not depend on the path, apart from a constant.
Surface and Volume Integrals
Just as with curves, it is possible to parameterise surfaces then integrate over those parameters without regard to geometry of the surface.
That is, to integrate a scalar function V over a surface A parameterised by r and s we calculate
where J is the Jacobian of the transformation to the parameters.
To integrate a vector this way, we integrate each component separately.
However, in three dimensions, every surface has an associated normal vector n, which can be used in integration. We write dS=ndS.
For a scalar function, V, and a vector function, v, this gives us the integrals
These integrals can be reduced to parametric integrals but, written this way, it is clear that they reflect more of the geometry of the surface.
When working in three dimensions, dV is a scalar, so there is only one option for integrals over volumes.
Gauss's divergence theorem
We know that, in one dimension,
Integration is the inverse of differentiation, so integrating the differential of a function returns the original function.
This can be extended to two or more dimensions in a natural way, drawing on the analogies between single variable and multivariable calculus.
The analog of D is ∇, so we should consider cases where the integrand is a divergence.
Instead of integrating over a one-dimensional interval, we need to integrate over a n-dimensional volume.
In one dimension, the integral depends on the values at the edges of the interval, so we expect the result to be connected with values on the boundary.
This suggests a theorem of the form,
This is indeed true, for vector fields in any number of dimensions.
This is called Gauss's theorem.
There are two other, closely related, theorems for grad and curl:
,
,
with the last theorem only being valid where curl is defined.
Stokes' curl theorem
These theorems also hold in two dimensions, where they relate surface and line integrals. Gauss's divergence theorem becomes
where s is arclength along the boundary curve and the vector n is the unit normal to the curve that lies in the surface S, i.e. in the tangent plane of the surface at its boundary, which is not necessarily the same as the unit normal associated with the boundary curve itself.
Similarly, we get
,
where C is the boundary of S
In this case the integral does not depend on the surface S.
To see this, suppose we have different surfaces, S1 and S2, spanning the same curve C, then by switching the direction of the normal on one of the surfaces we can write
The left hand side is an integral over a closed surface bounding some volume V so we can use Gauss's divergence theorem.
but we know this integrand is always zero so the right hand side of (2) must always be zero, i.e. the integral is independent of the surface.
This means we can choose the surface so that the normal to the curve lying in the surface is the same as the curves intrinsic normal.
Then, if u itself lies in the surface, we can write
just as we did for line integrals in the plane earlier, and substitute this into (1) to get
This is Stokes' curl theorem
Ordinary Differential Equations
Ordinary differential equations involve equations containing:
- variables
- functions
- their derivatives
and their solutions.
In studying integration, you already have considered solutions to very simple differential equations. For example, when you look to solving
for g(x), you are really solving the differential equation
Notations and terminology
The notations we use for solving differential equations will be crucial in the ease of solubility for these equations.
This document will be using three notations primarily:
- f' to denote the derivative of f
- D f to denote the derivative of f
to denote the derivative of f (for separable equations).
Terminology
Consider the differential equation
Since the equation's highest derivative is 2, we say that the differential equation is of order 2.
Some simple differential equations
A key idea in solving differential equations will be that of integration.
Let us consider the second order differential equation (remember that a function acts on a value).
How would we go about solving this? It tells us that on differentiating twice, we obtain the constant 2 so, if we integrate twice, we should obtain our result.
Integrating once first of all:
We have transformed the apparently difficult second order differential equation into a rather simpler one, viz.
This equation tells us that if we differentiate a function once, we get . If we integrate once more, we should find the solution.
This is the solution to the differential equation. We will get for all values of
and
.
The values and
are related to quantities known as initial conditions.
Why are initial conditions useful? ODEs (ordinary differential equations) are useful in modeling physical conditions. We may wish to model a certain physical system which is initially at rest (so one initial condition may be zero), or wound up to some point (so an initial condition may be nonzero, say 5 for instance) and we may wish to see how the system reacts under such an initial condition.
When we solve a system with given initial conditions, we substitute them after our process of integration.
Example
When we solved say we had the initial conditions
and
. (Note, initial conditions need not occur at f(0)).
After we integrate we make substitutions:
Without initial conditions, the answer we obtain is known as the general solution or the solution to the family of equations. With them, our solution is known as a specific solution.
Basic first order DEs
In this section we will consider four main types of differential equations:
- separable
- homogeneous
- linear
- exact
There are many other forms of differential equation, however, and these will be dealt with in the next section
Separable equations
A separable equation is in the form (using dy/dx notation which will serve us greatly here)
Previously we have only dealt with simple differential equations with g(y)=1. How do we solve such a separable equation as above?
We group x and dx terms together, and y and dy terms together as well.
Integrating both sides with respect to y on the left hand side and x on the right hand side:
we will obtain the solution.
Worked example
Here is a worked example illustrating the process.
We are asked to solve
Separating
Integrating
Letting where k is a constant we obtain
which is the general solution.
Verification
This step does not need to be part of your work, but if you want to check your solution, you can verify your answer by differentiation.
We obtained
as the solution to
Differentiating our solution with respect to x,
And since , we can write
We see that we obtain our original differential equation, thus our work must be correct.
Homogeneous equations
A homogeneous equation is in the form
This looks difficult as it stands, however we can utilize the substitution
so that we are now dealing with F(v) rather than F(y/x).
Now we can express y in terms of v, as y=xv and use the product rule.
The equation above then becomes, using the product rule
Then
which is a separable equation and can be solved as above.
However let's look at a worked equation to see how homogeneous equations are solved.
Worked example
We have the equation
This does not appear to be immediately separable, but let us expand to get
Substituting y=xv which is the same as substituting v=y/x:
Now
Canceling v from both sides
Separating
Integrating both sides
which is our desired solution.
Linear equations
A linear first order differential equation is a differential equation in the form
Multiplying or dividing this equation by any non-zero function of x makes no difference to its solutions so we could always divide by a(x) to make the coefficient of the differential 1, but writing the equation in this more general form may offer insights.
At first glance, it is not possible to integrate the left hand side, but there is one special case. If b happens to be the differential of a then we can write
and integration is now straightforward.
Since we can freely multiply by any function, lets see if we can use this freedom to write the left hand side in this special form.
We multiply the entire equation by an arbitrary, I(x), getting
then impose the condition
If this is satisfied the new left hand side will have the special form. Note that multiplying I by any constant will leave this condition still satisfied.
Rearranging this condition gives
We can integrate this to get
We can set the constant k to be 1, since this makes no difference.
Next we use I on the original differential equation, getting
Because we've chosen I to put the left hand side in the special form we can rewrite this as
Integrating both sides and dividing by I we obtain the final result
We call I an integrating factor. Similar techniques can be used on some other calculus problems.
Example
Consider
First we calculate the integrating factor.
Multiplying the equation by this gives
or
We can now integrate
Exact equations
An exact equation is in the form
- f(x, y) dx + g(x, y) dy = 0
and, has the property that
- Dx f = Dy g
(If the differential equation does not have this property then we can't proceed any further).
As a result of this, if we have an exact equation then there exists a function h(x, y) such that
- Dy h = f and Dx h = g
So then the solutions are in the form
- h(x, y) = c
by using the fact of the total differential. We can find then h(x, y) by integration
Basic second and higher order ODE's
The generic solution of a nth order ODE will contain n constants of integration. To calculate them we need n more equations. Most often, we have either
- boundary conditions, the values of y and its derivatives take for two different values of x
or
- initial conditions, the values of y and its first n-1 derivatives take for one particular value of x.
Reducible ODE's
1. If the independent variable, x, does not occur in the differential equation then its order can be lowered by one. This will reduce a second order ODE to first order.
Consider the equation:
Define
Then
Substitute these two expression into the equation and we get
=0
which is a first order ODE
Example
Solve
if at x=0, y=Dy=1
First, we make the substitution, getting
This is a first order ODE. By rearranging terms we can separate the variables
Integrating this gives
We know the values of y and u when x=0 so we can find c
Next, we reverse the substitution
and take the square root
To find out which sign of the square root to keep, we use the initial condition, Dy=1 at x=0, again, and rule out the negative square root. We now have another separable first order ODE,
Its solution is
Since y=1 when x=0, d=2/3, and
2. If the dependent variable, y, does not occur in the differential equation then it may also be reduced to a first order equation.
Consider the equation:
Define
Then
Substitute these two expressions into the first equation and we get
=0
which is a first order ODE
Linear ODEs
An ODE of the form
is called linear. Such equations are much simpler to solve than typical non-linear ODEs. Though only a few special cases can be solved exactly in terms of elementary functions, there is much that can be said about the solution of a generic linear ODE. A full account would be beyond the scope of this book
If F(x)=0 for all x the ODE is called homogeneous
Two useful properties of generic linear equations are
- Any linear combination of solutions of an homogeneous linear equation is also a solution.
- If we have a solution of a nonhomogeneous linear equation and we add any solution of the corresponding homogenous linear equation we get another solution of the nonhomogeneous linear equation
Variation of constants
Suppose we have a linear ODE,
and we know one solution, y=w(x)
The other solutions can always be written as y=wz. This substitution in the ODE will give us terms involving every differential of z upto the nth, no higher, so we'll end up with an nth order linear ODE for z.
We know that z is constant is one solution, so the ODE for z must not contain a z term, which means it will effectively be an n-1th order linear ODE. We will have reduced the order by one.
Lets see how this works in practice.
Example
Consider
One solution of this is y=x2, so substitute y=zx2 into this equation.
Rearrange and simplify.
This is first order for Dz. We can solve it to get
Since the equation is linear we can add this to any multiple of the other solution to get the general solution,
Linear homogeneous ODE's with constant coefficients
Suppose we have a ODE
we can take an inspired guess at a solution (motivate this)
For this function Dny=pny so the ODE becomes
y=0 is a trivial solution of the ODE so we can discard it. We are then left with the equation
This is called the characteristic equation of the ODE.
It can have up to n roots, p1, p2 … pn, each root giving us a different solution of the ODE.
Because the ODE is linear, we can add all those solution together in any linear combination to get a general solution
To see how this works in practice we will look at the second order case. Solving equations like this of higher order uses exactly the same principles; only the algebra is more complex.
Second order
If the ODE is second order,
then the characteristic equation is a quadratic,
with roots
What these roots are like depends on the sign of b2-4c, so we have three cases to consider.
1) b2 > 4c
In this case we have two different real roots, so we can write down the solution straight away.
2) b2 < 4c
In this case, both roots are imaginary. We could just put them directly in the formula, but if we are interested in real solutions it is more useful to write them another way.
Defining k2=4c-b2, then the solution is
For this to be real, the A's must be complex conjugates
Make this substitution and we can write,
If b is positive, this is a damped oscillation.
3) b2 = 4c
In this case the characteristic equation only gives us one root, p=-b/2. We must use another method to find the other solution.
We'll use the method of variation of constants. The ODE we need to solve is,
rewriting b and c in terms of the root. From the characteristic equation we know one solution is so we make the substitution
, giving
This simplifies to D2z=0, which is easily solved. We get
so the second solution is the first multiplied by x.
Higher order linear constant coefficient ODE's behave similarly: an exponential for every real root of the characteristic and a exponent multiplied by a trig factor for every complex conjugate pair, both being multiplied by a polynomial if the root is repeated.
E.g., if the characteristic equation factors to
the general solution of the ODE will be
The most difficult part is finding the roots of the characteristic equation.
Linear nonhomogeneous ODEs with constant coefficients
First, let's consider the ODE
a nonhomogeneous first order ODE which we know how to solve.
Using the integrating factor e-x we find
This is the sum of a solution of the corresponding homogeneous equation, and a polynomial.
Nonhomogeneous ODE's of higher order behave similarly.
If we have a single solution, yp of the nonhomogeneous ODE, called a particular solution,
then the general solution is y=yp+yh, where yh is the general solution of the homogeneous ODE.
Find yp for an arbitrary F(x) requires methods beyond the scope of this chapter, but there are some special cases where finding yp is straightforward.
Remember that in the first order problem yp for a polynomial F(x) was itself a polynomial of the same order. We can extend this to higher orders.
Example:
Consider a particular solution
Substitute for y and collect coefficients
So b2=0, b1=-7, b0=1, and the general solution is
This works because all the derivatives of a polynomial are themselves polynomials.
Two other special cases are
where Pn,Qn,An, and Bn are all polynomials of degree n.
Making these substitutions will give a set of simultaneous linear equations for the coefficients of the polynomials.
Partial Differential Equations
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Introduction
First order
Any partial differential equation of the form
where h1, h2 … hn, and b are all functions of both u and Rn can be reduced to a set of ordinary differential equations.
To see how to do this, we will first consider some simpler problems.
Special cases
We will start with the simple PDE
Because u is only differentiated with respect to z, for any fixed x and y we can treat this like the ODE, du/dz=u. The solution of that ODE is cez, where c is the value of u when z=0, for the fixed x and y
Therefore, the solution of the PDE is
Instead of just having a constant of integration, we have an arbitrary function. This will be true for any PDE.
Notice the shape of the solution, an arbitrary function of points in the xy, plane, which is normal to the 'z' axis, and the solution of an ODE in the 'z' direction.
Now consider the slightly more complex PDE
where h can be any function, and each a is a real constant.
We recognize the left hand side as being a·∇, so this equation says that the differential of u in the a direction is h(u). Comparing this with the first equation suggests that the solution can be written as an arbitrary function on the plane normal to a combined with the solution of an ODE.
Remembering from Calculus/Vectors that any vector r can be split up into components parallel and perpendicular to a,
we will use this to split the components of r in a way suggested by the analogy with (1).
Let's write
and substitute this into (2), using the chain rule. Because we are only differentiating in the a direction, adding any function of the perpendicular vector to s will make no difference.
First we calculate grad s, for use in the chain rule,
On making the substitution into (2), we get,
which is an ordinary differential equation with the solution
The constant c can depend on the perpendicular components, but not upon the parallel coordinate. Replacing s with a monotonic scalar function of s multiplies the ODE by a function of s, which doesn't affect the solution.
Example:
For this equation, a is (1, -1), s=x-t, and the perpendicular vector is (x+t)(1, 1). The reduced ODE is du/ds=0 so the solution is
- u=f(x+t)
To find f we need initial conditions on u. Are there any constraints on what initial conditions are suitable?
Consider, if we are given
- u(x,0), this is exactly f(x),
- u(3t,t), this is f(4t) and f(t) follows immediately
- u(t3+2t,t), this is f(t3+3t) and f(t) follows, on solving the cubic.
- u(-t,t), then this is f(0), so if the given function isn't constant we have a inconsistency, and if it is the solution isn't specified off the initial line.
Similarly, if we are given u on any curve which the lines x+t=c intersect only once, and to which they are not tangent, we can deduce f.
For any first order PDE with constant coefficients, the same will be true. We will have a set of lines, parallel to r=at, along which the solution is gained by integrating an ODE with initial conditions specified on some surface to which the lines aren't tangent.
If we look at how this works, we'll see we haven't actually used the constancy of a, so let's drop that assumption and look for a similar solution.
The important point was that the solution was of the form u=f(x(s),y(s)), where (x(s),y(s)) is the curve we integrated along -- a straight line in the previous case. We can add constant functions of integration to s without changing this form.
Consider a PDE,
For the suggested solution, u=f(x(s),y(s)), the chain rule gives
Comparing coefficients then gives
so we've reduced our original PDE to a set of simultaneous ODE's. This procedure can be reversed.
The curves (x(s),y(s)) are called characteristics of the equation.
Example: Solve given u=f(x) for x≥0 The ODE's are
subject to the initial conditions at s=0,
This ODE is easily solved, giving
so the characteristics are concentric circles round the origin, and in polar coordinates u(r,θ)=f(r)
Considering the logic of this method, we see that the independence of a and b from u has not been used either, so that assumption too can be dropped, giving the general method for equations of this quasilinear form.
Quasilinear
Summarising the conclusions of the last section, to solve a PDE
subject to the initial condition that on the surface, (x1(r1,…,rn-1, …xn(r1,…,rn-1), u=f(r1,…,rn-1) --this being an arbitrary paremetrisation of the initial surface--
- we transform the equation to the equivalent set of ODEs,
- subject to the initial conditions
- Solve the ODE's, giving xi as a function of s and the ri.
- Invert this to get s and the ri as functions of the xi.
- Substitute these inverse functions into the expression for u as a function of s and the ri obtained in the second step.
Both the second and third steps may be troublesome.
The set of ODEs is generally non-linear and without analytical solution. It may even be easier to work with the PDE than with the ODEs.
In the third step, the ri together with s form a coordinate system adapted for the PDE. We can only make the inversion at all if the Jacobian of the transformation to Cartesian coordinates is not zero,
This is equivalent to saying that the vector (a1, &hellip:, an) is never in the tangent plane to a surface of constant s.
If this condition is not false when s=0 it may become so as the equations are integrated. We will soon consider ways of dealing with the problems this can cause.
Even when it is technically possible to invert the algebraic equations it is obviously inconvenient to do so.
Example
To see how this works in practice, we will
a/ consider the PDE,
with generic initial condition,
Naming variables for future convenience, the corresponding ODE's are
subject to the initial conditions at τ=0
These ODE's are easily solved to give
These are the parametric equations of a set of straight lines, the characteristics.
The determinant of the Jacobian of this coordinate transformation is
This determinant is 1 when t=0, but if fr is anywhere negative this determinant will eventually be zero, and this solution fails.
In this case, the failure is because the surface is an envelope of the characteristics.
For arbitrary f we can invert the transformation and obtain an implicit expression for u
If f is given this can be solved for u.
1/ , The implicit solution is
This is a line in the u-x plane, rotating clockwise as t increases. If a is negative, this line eventually become vertical. If a is positive, this line tends towards u=0, and the solution is valid for all t.
2/ f(x,y)=x2, The implicit solution is
This solution clearly fails when , which is just when
. For any t>0 this happens somewhere. As t increases this point of failure moves toward the origin.
Notice that the point where u=0 stays fixed. This is true for any solution of this equation, whatever f is.
We will see later that we can find a solution after this time, if we consider discontinuous solutions. We can think of this as a shockwave.
3/
The implicit solution is
and we can not solve this explitely for u. The best we can manage is a numerical solution of this equation.
b/We can also consider the closely related PDE
The corresponding ODE's are
subject to the initial conditions at τ=0
These ODE's are easily solved to give
Writing f in terms of u, s, and τ, then substituting into the equation for x gives an implicit solution
It is possible to solve this for u in some special cases, but in general we can only solve this equation numerically. However, we can learn much about the global properties of the solution from further analysis
Characteristic initial value problems
What if initial conditions are given on a characteristic, on an envelope of characteristics, on a surface with characteristic tangents at isolated points?
Discontinuous solutions
So far, we've only considered smooth solutions of the PDE, but this is too restrictive. We may encounter initial conditions which aren't smooth, e.g.
If we were to simply use the general solution of this equation for smooth initial conditions,
we would get
which appears to be a solution to the original equation. However, since the partial differentials are undefined on the characteristic x+ct=0, so it becomes unclear what it means to say that the equation is true at that point.
We need to investigate further, starting by considering the possible types of discontinuities.
If we look at the derivations above, we see we've never use any second or higher order derivatives so it doesn't matter if they aren't continuous, the results above will still apply.
The next simplest case is when the function is continuous, but the first derivative is not, e.g. |x|. We'll initially restrict ourselves to the two-dimensional case, u(x, t) for the generic equation.
Typically, the discontinuity is not confined to a single point, but is shared by all points on some curve, (x0(s), t0(s) )
Then we have
We can then compare u and its derivatives on both sides of this curve.
It will prove useful to name the jumps across the discontinuity. We say
Now, since the equation (1) is true on both sides of the discontinuity, we can see that both u+ and u-, being the limits of solutions, must themselves satisfy the equation. That is,
Subtracting then gives us an equation for the jumps in the differentials
We are considering the case where u itself is continuous so we know that [u]=0. Differentiating this with respect to s will give us a second equation in the differential jumps.
The last two equations can only be both true if one is a multiple of the other, but multiplying s by a constant also multiplies the second equation by that same constant while leaving the curve of discontinuity unchanged, hence we can without loss of generality define s to be such that
But these are the equations for a characteristic, i.e. discontinuities propagate along characteristics. We could use this property as an alternative definition of characteristics.
We can deal similarly with discontinuous functions by first writing the equation in conservation form, so called because conservation laws can always be written this way.
Notice that the left hand side can be regarded as the divergence of (au, bu). Writing the equation this way allows us to use the theorems of vector calculus.
Consider a narrow strip with sides parallel to the discontinuity and width h
We can integrate both sides of (1) over R, giving
Next we use Green's theorem to convert the left hand side into a line integral.
Now we let the width of the strip fall to zero. The right hand side also tends to zero but the left hand side reduces to the difference between two integrals along the part of the boundary of R parallel to the curve.
The integrals along the opposite sides of R have different signs because they are in opposite directions.
For the last equation to always be true, the integrand must always be zero, i.e.
Since, by assumption [u] isn't zero, the other factor must be, which immediately implies the curve of discontinuity is a characteristic.
Once again, discontinuities propagate along characteristics.
Above, we only considered functions of two variables, but it is straightforward to extend this to functions of n variables.
The initial condition is given on an n-1 dimensional surface, which evolves along the characteristics. Typical discontinuities in the initial condition will lie on a n-2 dimensional surface embedded within the initial surface. This surface of discontinuity will propagate along the characteristics that pass through the initial discontinuity.
The jumps themselves obey ordinary differential equations, much as u itself does on a characteristic. In the two dimensional case, for u continuous but not smooth, a little algebra shows that
while u obeys the same equation as before,
We can integrate these equations to see how the discontinuity evolves as we move along the characteristic.
We may find that, for some future s, [ux] passes through zero. At such points, the discontinuity has vanished, and we can treat the function as smooth at that characteristic from then on.
Conversely, we can expect that smooth functions may, under the righr circumstances, become discontinuous.
To see how all this works in practice we'll consider the solutions of the equation
for three different initial conditions.
The general solution, using the techniques outlined earlier, is
u is constant on the characteristics, which are straight lines with slope dependent on u.
First consider f such that
While u is continuous its derivative is discontinuous at x=0, where u=0, and at x=a, where u=1. The characteristics through these points divide the solution into three regions.
All the characteristics to the right of the characteristic through x=a, t=0 intersect the x-axis to the right of x=1, where u=1 so u is 1 on all those characteristics, i.e whenever x-t>a.
Similarly the characteristic through the origin is the line x=0, to the left of which u remains zero.
We could find the value of u at a point in between those two characteristics either by finding which intermediate characteristic it lies on and tracing it back to the initial line, or via the general solution.
Either way, we get
At larger t the solution u is more spread out than at t=0 but still the same shape.
We can also consider what happens when a tends to 0, so that u itself is discontinuous at x=0.
If we write the PDE in conservation form then use Green's theorem, as we did above for the linear case, we get
[u²] is the difference of two squares, so if we take s=t we get
In this case the discontinuity behaves as if the value of u on it were the average of the limiting values on either side.
However, there is a caveat.
Since the limiting value to the left is u- the discontinuity must lie on that characteristic, and similarly for u+; i.e the jump discontinuity must be on an intersection of characteristics, at a point where u would otherwise be multivalued.
For this PDE the characteristic can only intersect on the discontinuity if
If this is not true the discontinuity can not propagate. Something else must happen.
The limit a=0 is an example of a jump discontinuity for which this condition is false, so we can see what happens in such cases by studying it.
Taking the limit of the solution derived above gives
If we had taken the limit of any other sequence of initials conditions tending to the same limit we would have obtained a trivially equivalent result.
Looking at the characteristics of this solution, we see that at the jump discontinuity characteristics on which u takes every value betweeen 0 and 1 all intersect.
At later times, there are two slope discontinuities, at x=0 and x=t, but no jump discontinuity.
This behaviour is typical in such cases. The jump discontinuity becomes a pair of slope discontinuities between which the solution takes all appropriate values.
Now, lets consider the same equation with the initial condition
This has slope discontinuities at x=0 and x=a, dividing the solution into three regions.
The boundaries between these regions are given by the characteristics through these initial points, namely the two lines
These characteristics intersect at t=a, so the nature of the solution must change then.
In between these two discontinuities, the characteristic through x=b at t=0 is clearly
All these characteristics intersect at the same point, (x,t)=(a,a).
We can use these characteristics, or the general solution, to write u for t<a
As t tends to a, this becomes a step function. Since u is greater to the left than the right of the discontinuity, it meets the condition for propagation deduced above, so for t>a u is a step function moving at the average speed of the two sides.
This is the reverse of what we saw for the initial condition previously considered, two slope discontinuities merging into a step discontinuity rather than vice versa. Which actually happens depends entirely on the initial conditions. Indeed, examples could be given for which both processes happen.
In the two examples above, we started with a discontinuity and investigated how it evolved. It is also possible for solutions which are initially smooth to become discontinuous.
For example, we saw earlier for this particular PDE that the solution with the initial condition u=x² breaks down when 2xt+1=0. At these points the solution becomes discontinuous.
Typically, discontinuities in the solution of any partial differential equation, not merely ones of first order, arise when solutions break down in this way and propagate similarly, merging and splitting in the same fashion.
Fully non-linear PDEs
It is possible to extend the approach of the previous sections to reduce any equation of the form
to a set of ODE's, for any function, F.
We will not prove this here, but the corresponding ODE's are
If u is given on a surface parameterized by r1…rn then we have, as before, n initial conditions on the n, xi
given by the parameterization and one initial condition on u itself,
but, because we have an extra n ODEs for the ui's, we need an extra n initial conditions.
These are, n-1 consistency conditions,
which state that the ui's are the partial derivatives of u on the initial surface, and one initial condition
stating that the PDE itself holds on the initial surface.
These n initial conditions for the ui will be a set of algebraic equations, which may have multiple solutions. Each solution will give a different solution of the PDE.
Example
Consider
The initial conditions at τ=0 are
and the ODE's are
Note that the partial derivatives are constant on the characteristics. This always happen when the PDE contains only partial derivatives, simplifying the procedure.
These equations are readily solved to give
On eliminating the parameters we get the solution,
which can easily be checked. abc
Second order
Suppose we are given a second order linear PDE to solve
The natural approach, after our experience with ordinary differential equations and with simple algebraic equations, is attempt a factorisation. Let's see how for this takes us.
We would expect factoring the left hand of (1) to give us an equivalent equation of the form
and we can immediately divide through by a. This suggests that those particular combinations of first order derivatives will play a special role.
Now, when studying first order PDE's we saw that such combinations were equivalent to the derivatives along characteristic curves. Effectively, we changed to a coordinate system defined by the characteristic curve and the initial curve.
Here, we have two combinations of first order derivatives each of which may define a different characteristic curve. If so, the two sets of characteristics will define a natural coordinate system for the problem, much as in the first order case.
In the new coordinates we will have
with each of the factors having become a differentiation along its respective characteristic curve, and the left hand side will become simply urs giving us an equation of the form
If A, B, and C all happen to be zero, the solution is obvious. If not, we can hope that the simpler form of the left hand side will enable us to make progress.
However, before we can do all this, we must see if (1) can actually be factored.
Multiplying out the factors gives
On comparing coefficients, and solving for the α's we see that they are the roots of
Since we are discussing real functions, we are only interested in real roots, so the existence of the desired factorization will depend on the discriminant of this quadratic equation.
- then we have two factors, and can follow the procedure outlined above. Equations like this are called hyperbolic
- then we have only factor, giving us a single characteristic curve. It will be natural to use distance along these curves as one coordinate, but the second must be determined by other considerations.
- The same line of argument as before shows that use the characteristic curve this way gives a second order term of the form urr, where we've only taken the second derivative with respect to one of the two coordinates. Equations like this are called parabolic
- then we have no real factors. In this case the best we can do is reduce the second order terms to the simplest possible form satisfying this inequality, i.e urr+uss
- It can be shown that this reduction is always possible. Equations like this are called elliptic
It can be shown that, just as for first order PDEs, discontinuities propagate along characteristics. Since elliptic equations have no real characteristics, this implies that any discontinuities they may have will be restricted to isolated points; i.e., that the solution is almost everywhere smooth.
This is not true for hyperbolic equations. Their behavior is largely controlled by the shape of their characteristic curves.
These differences mean different methods are required to study the three types of second equation. Fortunately, changing variables as indicated by the factorisation above lets us reduce any second order PDE to one in which the coefficients of the second order terms are constant, which means it is sufficient to consider only three standard equations.
We could also consider the cases where the right hand side of these equations is a given function, or proportional to u or to one of its first order derivatives, but all the essential properties of hyperbolic, parabolic, and elliptic equations are demonstrated by these three standard forms.
While we've only demonstrated the reduction in two dimensions, a similar reduction applies in higher dimensions, leading to a similar classification. We get, as the reduced form of the second order terms,
where each of the ais is equal to either 0, +1, or -1.
If all the ais have the same sign the equation is elliptic
If any of the ais are zero the equation is parabolic
If exactly one of the ais has the opposite sign to the rest the equation is hyperbolic
In 2 or 3 dimensions these are the only possibilities, but in 4 or more dimensions there is a fourth possibility: at least two of the ais are positive, and at least two of the ais are negative.
Such equations are called ultrahyperbolic. They are less commonly encountered than the other three types, so will not be studied here.
When the coefficients are not constant, an equation can be hyperbolic in some regions of the xy plane, and elliptic in others. If so, different methods must be used for the solutions in the two regions.
Elliptic
Standard form, Laplace's equation:
Quote equation in spherical and cylindrical coordinates, and give full solution for cartesian and cylindrical coordinates. Note averaging property Comment on physical significance, rotation invariance of laplacian.
Hyperbolic
Standard form, wave equation:
Solution, any sum of functions of the form
These are waves. Compare with solution from separating variables. Domain of dependence, etc.
Parabolic
The canonical parabolic equation is the diffusion equation:
Here, we will consider some simple solutions of the one-dimensional case.
The properties of this equation are in many respects intermediate between those of hyperbolic and elliptic equation.
As with hyperbolic equations but not elliptic, the solution is well behaved if the value is given on the initial surface t=0.
However, the characteristic surfaces of this equation are the surfaces of constant t, thus there is no way for discontinuities to propagate to positive t.
Therefore, as with elliptic equations but not hyberbolic, the solutions are typically smooth, even when the initial conditions aren't.
Furthermore, at a local maximum of h, its Laplacian is negative, so h is decreasing with t, while at local minima, where the Laplacian will be positive, h will increase with t. Thus, initial variations in h will be smoothed out as t increases.
In one dimension, we can learn more by integrating both sides,
Provided that hx tends to zero for large x, we can take the limit as a and b tend to infinity, deducing
so the integral of h over all space is constant.
This means this PDE can be thought of as describing some conserved quantity, initially concentrated but spreading out, or diffusing, over time.
This last result can be extended to two or more dimensions, using the theorems of vector calculus.
We can also differentiate any solution with respect to any coordinate to obtain another solution. E.g. if h is a solution then
so hx also satisfies the diffusion equation.
Similarity solution
Looking at this equation, we might notice that if we make the change of variables
then the equation retains the same form. This suggests that the combination of variables x²/t, which is unaffected by this variable change, may be significant.
We therefore assume this equation to have a solution of the special form
then
and substituting into the diffusion equation eventually gives
which is an ordinary differential equation.
Integrating once gives
Reverting to h, we find
This last integral can not be written in terms of elementary functions, but its values are well known.
In particular the limiting values of h at infinity are
taking the limit as t tends to zero gives
We see that the initial discontinuity is immediately smoothed out. The solution at later times retains the same shape, but is more stretched out.
The derivative of this solution with respect to x
is itself a solution, with h spreading out from its initial peak, and plays a significant role in the further analysis of this equation.
The same similarity method can also be applied to some non-linear equations.
Separating variables
We can also obtain some solutions of this equation by separating variables.
giving us the two ordinary differential equations
and solutions of the general form
<noinclude>
Exercises
Calculus/Multivariable and differential calculus:Solutions </noinclude>
Extensions
Systems of Ordinary Differential Equations
We have already examined cases where we have a single differential equation and found several methods to aid us in finding solutions to these equations. But what happens if we have two or more differential equations that depend on each other? For example, consider the case where
and
Such a set of differential equations is said to be coupled. Systems of ordinary differential equations such as these are what we will look into in this section.
First order systems
A general system of differential equations can be written in the form
Instead of writing the set of equations in a vector, we can write out each equation explicitly, in the form:
If we have the system at the very beginning, we can write it as:
where
and
or write each equation out as shown above.
Why are these forms important? Often, this arises as a single, higher order differential equation that is changed into a simpler form in a system. For example, with the same example,
we can write this as a higher order differential equation by simple substitution.
then
Notice now that the vector form of the system is dependent on t since
the first component is dependent on t. However, if instead we had
notice the vector field is no longer dependent on t. We call such systems autonomous. They appear in the form
We can convert between an autonomous system and a non-autonomous one by simply making a substitution that involves t, such as y=(x, t), to get a system:
In vector form, we may be able to separate F in a linear fashion to get something that looks like:
where A(t) is a matrix and b is a vector. The matrix could contain functions or constants, clearly, depending on whether the matrix depends on t or not.
Real numbers
Fields
You are probably already familiar with many different sets of numbers from your past experience. Some of the commonly used sets of numbers are
- Natural numbers, usually denoted with an N, are the numbers 0,1,2,3,...
- Integers, usually denoted with a Z, are the positive and negative natural numbers: ...-3,-2,-1,0,1,2,3...
- Rational numbers, denoted with a Q, are fractions of integers (excluding division by zero): -1/3, 5/1, 0, 2/7. etc.
- Real numbers, denoted with a R, are constructed and discussed below.
Note that different sets of numbers have different properties. In the set integers for example, any number always has an additive inverse: for any integer x, there is another integer t such that This should not be terribly surprising: from basic arithmetic we know that
. Try to prove to yourself that not all natural numbers have an additive inverse.
In mathematics, it is useful to note the important properties of each of these sets of numbers. The rational numbers, which will be of primary concern in constructing the real numbers, have the following properties:
- There exists a number 0 such that for any other number a, 0+a=a+0=a
- For any two numbers a and b, a+b is another number
- For any three numbers a,b, and c, a+(b+c)=(a+b)+c
- For any number a there is another number -a such that a+(-a)=0
- For any two numbers a and b, a+b=b+a
- For any two numbers a and b,a*b is another number
- There is a number 1 such that for any number a, a*1=1*a=a
- For any two numbers a and b, a*b=b*a
- For any three numbers a,b and c, a(bc)=(ab)c
- For any three numbers a,b and c, a(b+c)=ab+ac
- For every number a there is another number a-1 such that aa-1=1
As presented above, these may seem quite intimidating. However, these properties are nothing more than basic facts from arithmetic. Any collection of numbers (and operations + and * on those numbers) which satisfies the above properties is called a field. The properties above are usually called field axioms. As an exercise, determine if the integers form a field, and if not, which field axiom(s) they violate.
Even though the list of field axioms is quite extensive, it does not fully explore the properties of rational numbers. Rational numbers also have an ordering.' A total ordering must satisfy several properties: for any numbers a, b, and c
- if a ≤ b and b ≤ a then a = b (antisymmetry)
- if a ≤ b and b ≤ c then a ≤ c (transitivity)
- a ≤ b or b ≤ a (totality)
To familiarize yourself with these properties, try to show that (a) natural numbers, integers and rational numbers are all totally ordered and more generally (b) convince yourself that any collection of rational numbers are totally ordered (note that the integers and natural numbers are both collections of rational numbers).
Finally, it is useful to recognize one more characterization of the rational numbers: every rational number has a decimal expansion which is either repeating or terminating. The proof of this fact is omitted, however it follows from the definition of each rational number as a fraction. When performing long division, the remainder at any stage can only take on positive integer values smaller than the denominator, of which there are finitely many.
Constructing the Real Numbers
There are two additional tools which are needed for the construction of the real numbers: the upper bound and the least upper bound. Definition A collection of numbers E is bounded above if there exists a number m such that for all x in E x≤m. Any number m which satisfies this condition is called an upper bound of the set E.
Definition If a collection of numbers E is bounded above with m as an upper bound of E, and all other upper bounds of E are bigger than m, we call m the least upper bound or supremum of E, denoted by sup E.
Many collections of rational numbers do not have a least upper bound which is also rational, although some do. Suppose the numbers 5 and 10/3 are, together, taken to be E. The number 5 is not only an upper bound of E, it is a least upper bound. In general, there are many upper bounds (12, for instance, is an upper bound of the collection above), but there can be at most one least upper bound.
Consider the collection of numbers : You may recognize these decimals as the first few digits of pi. Since each decimal terminates, each number in this collection is a rational number. This collection has infinitely many upper bounds. The number 4, for instance, is an upper bound. There is no least upper bound, at least not in the rational numbers. Try to convince yourself of this fact by attempting to construct such a least upper bound: (a) why does pi not work as a least upper bound (hint: pi does not have a repeating or terminating decimal expansion), (b) what happens if the proposed supremum is equal to pi up to some decimal place, and zeros after (c) if the proposed supremum is bigger than pi, can you find a smaller upper bound which will work?
In fact, there are infinitely many collections of rational numbers which do not have a rational least upper bound. We define a real number to be any number that is the least upper bound of some collection of rational numbers.
Properties of Real Numbers
The reals are totally ordered.
- For all reals; a, b, c
- Either b>a, b=a, or b<a.
- If a<b and b<c then a<c
Also
- b>a implies b+c>a+c
- b>a and c>0 implies bc>ac
- b>a implies -a>-b
Upper bound axiom
- Every non-empty set of real numbers which is bounded above has a supremum.
The upper bound axiom is necessary for calculus. It is not true for rational numbers.
We can also define lower bounds in the same way.
Definition A set E is bounded below if there exists a real M such that for all x∈E x≥M Any M which satisfies this condition is called an lower bound of the set E
Definition If a set, E, is bounded below, M is an lower bound of E, and all other lower bounds of E are less than M, we call M the greatest lower bound or inifimum of E, denoted by inf E
The supremum and infimum of finite sets are the same as their maximum and minimum.
Theorem
- Every non-empty set of real numbers which is bounded below has an infimum.
Proof:
- Let E be a non-empty set of of real numbers, bounded below
- Let L be the set of all lower bounds of E
- L is not empty, by definition of bounded below
- Every element of E is an upper bound to the set L, by definition
- Therefore, L is a non empty set which is bounded above
- L has a supremum, by the upper bound axiom
- 1/ Every lower bound of E is ≤sup L, by definition of supremum
- Suppose there were an e∈E such that e<sup L
- Every element of L is ≤e, by definition
- Therefore e is an upper bound of L and e<sup L
- This contradicts the definition of supremum, so there can be no such e.
- If e∈E then e≥sup L, proved by contradiction
- 2/ Therefore, sup L is a lower bound of E
- inf E exists, and is equal to sup L, on comparing definition of infinum to lines 1 & 2
Bounds and inequalities, theorems:
Theorem: (The triangle inequality)
Proof by considering cases
If a≤b≤c then |a-c|+|c-b| = (c-a)+(c-b) = 2(c-b)+(b-a)>b-a = |b-a|
Exercise: Prove the other five cases.
This theorem is a special case of the triangle inequality theorem from geometry: The sum of two sides of a triangle is greater than or equal to the third side. It is useful whenever we need to manipulate inequalities and absolute values.
Complex Numbers
In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. Real numbers may be considered to be complex numbers with an imaginary part of zero; that is, the real number a is equivalent to the complex number a+0i.
For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + bi, the real part (a) is denoted Re(z), or ℜ(z), and the imaginary part (b) is denoted Im(z), or ℑ(z).
Complex numbers can be added, subtracted, multiplied, and divided like real numbers and have other elegant properties. For example, real numbers alone do not provide a solution for every polynomial algebraic equation with real coefficients, while complex numbers do (this is the fundamental theorem of algebra).
Equality
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di if and only if a = c and b = d.
Notation and operations
The set of all complex numbers is usually denoted by C, or in blackboard bold by (Unicode ℂ). The real numbers, R, may be regarded as "lying in" C by considering every real number as a complex: a = a + 0i.
Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i2 = −1:
Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.
In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.
The field of complex numbers
Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:
So defined, the complex numbers form a field, the complex number field, denoted by C (a field is an algebraic structure in which addition, subtraction, multiplication, and division are defined and satisfy certain algebraic laws. For example, the real numbers form a field).
The real number a is identified with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i can then be defined as the complex number (0, 1), which verifies
In C, we have:
- additive identity ("zero"): (0, 0)
- multiplicative identity ("one"): (1, 0)
- additive inverse of (a,b): (−a, −b)
- multiplicative inverse (reciprocal) of non-zero (a, b):
Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane.
The complex plane
A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram . The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.
Polar form
Alternatively, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument of z. For r = 0 any value of φ describes the same number. To get a unique representation, a conventional choice is to set arg(0) = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is called the polar form of the complex number.
Conversion from the polar form to the Cartesian form
Conversion from the Cartesian form to the polar form
The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function. A formula that uses the arccos function requires fewer case differentiations:
Notation of the polar form
The notation of the polar form as
is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as
which is called exponential form.
Multiplication, division, exponentiation, and root extraction in the polar form
Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.
Using sum and difference identities its possible to obtain that
and that
Exponentiation with integer exponents; according to de Moivre's formula,
Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.
The addition of two complex numbers is just the addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.
Multiplication by i corresponds to a counter-clockwise rotation by 90° (π/2 radians). The geometric content of the equation i 2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.
All the roots of any number, real or complex, may be found with a simple algorithm. The nth roots are given by
for k = 0, 1, 2, …, n − 1, where represents the principal nth root of r.
Absolute value, conjugation and distance
The absolute value (or modulus or magnitude) of a complex number z = r eiφ is defined as |z| = r. Algebraically, if z = a + bi, then
One can check readily that the absolute value has three important properties:
if and only if
(triangle inequality)
for all complex numbers z and w. It then follows, for example, that and
. By defining the distance function d(z, w) = |z − w| we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.
The complex conjugate of the complex number z = a + bi is defined to be a − bi, written as or
. As seen in the figure,
is the "reflection" of z about the real axis. The following can be checked:
if and only if z is real
if z is non-zero.
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
That conjugation commutes with all the algebraic operations (and many functions; e.g. ) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function
is not complex-differentiable.
Complex fractions
We can divide a complex number (a + bi) by another complex number (c + di) ≠ 0 in two ways. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easily derived. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. The new denominator is a real number.
Matrix representation of complex numbers
While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form
where a and b are real numbers. The sum and product of two such matrices is again of this form. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as
which suggests that we should identify the real number 1 with the identity matrix
and the imaginary unit i with
a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.
The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix.
If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z.
If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.
Advanced Integration Techniques
Integration by Complexifying
This technique requires an understanding and recognition of complex numbers. Specifically Euler's formula:
Recognize, for example, that the real portion:
Given an integral of the general form:
We can complexify it:
With basic rules of exponents:
It can be proven that the "real portion" operator can be moved outside the integral:
The integral easily evaluates:
Multiplying and dividing by (1-2i):
Which can be rewritten as:
Applying Euler's forumula:
Expanding:
Taking the Real part of this expression:
So:
Appendix
Calculus/Choosing delta
- This page is an addendum to Calculus/Formal Definition of the Limit.
Recall the definition of a limit:
A number L is the limit of a function f(x) as x approaches c if and only if for all numbers ε > 0 there exists a number δ > 0 such that
whenever
.
In other words, given a number ε we must construct a number δ such that assuming
we can prove
;
moreover, this proof must work for all values of ε > 0.
Note: this definition is not constructive -- it does not tell you how to find the limit L, only how to check whether a particular value is indeed the limit. We use the informal definition of the limit, experience with similar problems, or theorems (L'Hopital's rule, for example), to determine the value, and then can prove the correctness of this value using the formal definition.
Example 1: Suppose we want to find the limit of f(x) = x + 5 as x approaches c = 9. We know that the limit L is 9+5=14, and desire to prove this.
We choose δ = ε (this will be explained later). Then, since we assume
we can show
,
which is what we wanted to prove.
We chose δ by working backwards from the formula we are trying to prove:
.
In this case, we desire to prove
,
given
,
so the easiest way to prove it is by choosing δ = ε. This example, however, is too easy to adequately explain how to choose δ in general. Lets try something harder:
Example 2: Prove that the limit of f(x) = x² - 9 as x approaches 2 is L = -5.
We want to prove that
given
.
We choose δ by working backwards. First, we need to rewrite the equation we want to prove using δ instead of x:
Note: we used the fact that |x + 2| < δ + 4, which can be proven with the triangle inequality.
Word of caution: the above series of equations is not a logical series of steps, and is not part of any proof, but is an informal technique used to help write the proof. We will select a value of δ so that the last equation is true, and then use the last equation to prove the equations above it in turn (which is what was meant earlier by working backwards).
Note: in the equations above, when δ was substituted for x, the sign < was replaced with =. This can be done (but is not necessary) because we are not told that |x-2| = δ, but rather |x-2| < δ. The justification for this becomes clear when the above equations are used in backwards order in the proof.
We can solve this last equation for δ using the quadratic formula:
Note: δ is always in terms of ε. A constant value of δ (e.g., δ = 0.5) will never work.
Now, we have a value of δ, and we can do our proof:
given
,
.
Here a few more examples of choosing δ; try to figure them out before reading the explanation.
Example 3: Prove that the limit of f(x) = sin(x)/x as x approaches 0 is L = 1.
Explanation:
Example 4: Prove that f(x) = 1/x has no limit as x approaches 0.
Example 5: Prove that
Solution: To do it, we'll look at two cases: and
. The
case is easy. First let's let
. That means we want the values chosen in the domain to map to
in the range. We want a delta such that
so let's choose
. The chosen
defines the interval
in our domain. This gets mapped to
in our range, which is contained in
. Notice that
doesn't depend on
. So for
, we widen the interval in the range that we are allowed to map onto, but our interval in the domain stays fixed and always maps to the same sub-interval in the range. So
works for any
.
Now suppose . We want a
such that
whenever
. So let's assume
and work backwards to find a suitable
:
Since , we have
. Since both numbers above are positive, we can take the (positive) square root of both extremes of the inequality:
The above equation represents the distance, either negative or positive, that x can vary from 2 and still be within of 4. We want to choose the smaller of the two extremes to construct our interval. It turns out that
for
, so choose
. As a sanity check, let's try with
.
which is approximately
At the extreme right of the domain, this gives
and
which is within 0.002 of 4.
Exercise Solutions
Algebra Solutions


Precalculus Cumulative Exercise Set Solutions
Convert to interval notation









This is equivalent to

State the following intervals using set notation
![[3,4] \,](../../../upload.wikimedia.org/math/7/5/4/754867828f4da2744549663cc47923aa.png)





Which one of the following is a true statement?

Let . Then
, and
Thus,
false

Using the same example as above, we have .
false

true
Evaluate the following expressions





![\sqrt[3]{\frac{27}{8}}](../../../upload.wikimedia.org/math/c/d/c/cdc339e9ed8f3192d2aa4919648a87fa.png)


![\frac{\sqrt{27}}{\sqrt[3]{9}}](../../../upload.wikimedia.org/math/9/d/5/9d5b26e7021395a068fba060e4a05634.png)
Simplify the following








Functions
52. Let .


,

The domain is ; the range is
,

No, since isn't one-to-one; for example,
.
53. Let ,
.
- a. Give formulae for

.

.

.

.

provided
. Note that 0 is not in the domain of
, since it's not in the domain of
, and you can't divide by something that doesn't exist!

. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression
either.

.

.


;
.


Yes; and
. Note that
and its inverse are the same.
As pictured, by the Vertical Line test, this graph represents a function.
55. Consider the following function

56. Consider the following function

57. Consider the following function

58. Consider the following function

Limit Solutions
Infinite Limits/Infinity is not a number Solutions
Write out an explanatory paragraph for the following limits that include . Remember that you will have to change any comparison of magnitude between a real number and
to a different phrase. In the second case, you will have to work out for yourself what the formula means.

This formula says that I can make the values of as close as I would like to 0, so long as I make x sufficiently large.

This formula says that you can make the sum as close as you would like to 2 by making
sufficiently large.
Limits Cumulative Exercise Set Solutions
Basic Limit Exercises

Since this is a polynomial, two can simply be plugged in. This results in

One-Sided Limits
Evaluate the following limits or state that the limit does not exist.

Factor as . In this form we can see that there is a removable discontinuity at x=0 and that the limit is


is defined if
, so the limit is

is not defined if
, so the limit does not exist.
Two-Sided Limits
Evaluate the following limits or state that the limit does not exist.


The limit does not exist.

The limit does not exist.







The limit does not exist.

As approaches
, the denominator will be a very small positive number, so the whole fraction will be a very large positive number. Thus, the limit is
.

As approaches
, the numerator goes to 5 and the denominator goes to 0. Depending on whether you approach
from the left or the right, the denominator will be either a very small negative number, or a very small positive number. So the limit from the left is
and the limit from the right is
. Thus, the limit does not exist.



The limit does not exist.





Notice that as approaches
, the numerator approaches
while the denominator approaches
. However, if you approach from below, the denominator is negative, and if you approach from above, the denominator is positive. So the limits from the left and right will be
and
respectively. Thus, the limit does not exist.
Limits to Infinity
Evaluate the following limits or state that the limit does not exist.

This rational function is bottom-heavy, so the limit is .

This rational function has evenly matched powers of in the numerator and denominator, so the limit will be the ratio of the coefficients, i.e.
.

Balanced powers in the numerator and denominator, so the limit is the ratio of the coefficients, i.e. .

This is a top-heavy rational function, where the exponent of the ratio of the leading terms is . Since it is even, the limit will be
.

Bottom-heavy rational function, so the limit is .

This is a rational function, as can be seen by writing it in the form . Since the powers of
in the numerator and denominator are evenly matched, the limit will be the ratio of the coefficients, i.e.
.

Bottom-heavy, so the limit is .

Evenly matched highest powers of in the numerator and denominator, so the limit will be the ratio of the corresponding coefficients, i.e.
.

Top-heavy rational function, where the exponent of the ratio of the leading terms is , so the limit is
.

Bottom-heavy, so the limit is .
Limits of Piecewise Functions
Evaluate the following limits or state that the limit does not exist.
37. Consider the function



Since the limits from the left and right don't match, the limit does not exist.
38. Consider the function





Since the left and right limits match, the overall limit is also .

39. Consider the function




Since the limits from the right and left match, the overall limit is . Note that in this case, the limit at 2 does not match the function value at 2, so the function is discontinuous at this point.
Differentiation Solutions
Differentiation Defined Solutions


The definition of the slope of at
is
Substituting in and
gives:







Since the limits from the left and the right at are not equal, the limit does not exist, so
is not differentiable at
.




- Find the derivatives of the following equations:



Chain Rule Solutions




First method:
Second method:
The two methods give the same answer.



Some Important Theorems Solutions
Rolle's Thoerem

Mean Value Theorem






1: Using the expression from the mean value theorem
insert values. Our chosen interval is . So, we have
2: By the Mean Value Theorem, we know that somewhere in the interval exists a point that has the same slope as that point. Thus, let us take the derivative to find this point .
Now, we know that the slope of the point is 4. So, the derivative at this point is 4. Thus,
. So

![[0,\pi]](../../../upload.wikimedia.org/math/e/1/8/e1868564b62b4e2f1c063321df289469.png)
1: We start with the expression:
so,
(Remember, sin(π) and sin(0) are both 0.)
2: Now that we have the slope of the line, we must find the point x = c that has the same slope. We must now get the derivative!
The cosine function is 0 at (where
is an integer). Remember, we are bound by the interval
, so
is the point
that satisfies the Mean Value Theorem.
Basics of Differentiation Cumulative Exercise Set Solutions
Find the Derivative by Definition









Prove the Constant Rule

![\frac{d}{dx}\left[cf(x)\right] = c \frac{d}{dx}\left[f(x)\right]](../../../upload.wikimedia.org/math/4/3/a/43a914dad1efe3423b921e4dd6924280.png)
Find the Derivative by Rules
Power Rule

![f(x) = 3\sqrt[3]{x}\,](../../../upload.wikimedia.org/math/c/b/b/cbb20f52c448cac11f99b9527d212b6e.png)




![f(x) = \frac{3}{x^4} - \sqrt[4]{x} + x \,](../../../upload.wikimedia.org/math/7/a/5/7a5e8965529063e1dcb09e2b4ebe0782.png)

![f(x) = \frac{1}{\sqrt[3]{x}} + \sqrt{x} \,](../../../upload.wikimedia.org/math/a/1/9/a19701b4473e0092c7007aa739181d46.png)
Product Rule




Quotient Rule







Chain Rule

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then

Let . Then
Exponentials


Let . Then

Let
Then
Using the chain rule, we have
The individual factor are
So

Logarithms



Let . Then


Trigonometric functions


More Differentiation
![\frac{d}{dx}[(x^{3}+5)^{10}]](../../../upload.wikimedia.org/math/e/a/d/ead360cd62636e2512acbe4d656b2655.png)
![\frac{d}{dx}[x^{3}+3x]](../../../upload.wikimedia.org/math/d/b/c/dbce3ac485fac127f932d7a23e69537f.png)
![\frac{d}{dx}[(x+4)(x+2)(x-3)]](../../../upload.wikimedia.org/math/b/6/9/b69873a920a7a9f943285e11742f9add.png)
Let . Then
![\frac{d}{dx}[\frac{x+1}{3x^{2}}]](../../../upload.wikimedia.org/math/f/c/c/fcc667fbf7a0d6a3f6df15a53b8419bd.png)
![\frac{d}{dx}[3x^{3}]](../../../upload.wikimedia.org/math/6/1/9/619bbc93a4fb50b1c46c8fd61b88c25d.png)
![\frac{d}{dx}[x^{4}\sin x]](../../../upload.wikimedia.org/math/6/b/8/6b808ac8534dbed2d1eb509f63ee7340.png)

![\frac{d}{dx}[e^{x^{2}}]](../../../upload.wikimedia.org/math/b/8/c/b8c0a852488e802d4d9094c91b00cfe4.png)
![\frac{d}{dx}[e^{2^{x}}]](../../../upload.wikimedia.org/math/1/a/5/1a52029c7d3bbc2eb45db4d453f1f0de.png)
Implicit Differentiation
Use implicit differentiation to find y'


Logarithmic Differentiation
Use logarithmic differentiation to find :
![y = x(\sqrt[4]{1-x^3}\,)](../../../upload.wikimedia.org/math/c/a/6/ca6f41c9f6005c02d1bed079be037495.png)




Equation of Tangent Line
For each function, , (a) determine for what values of
the tangent line to
is horizontal and (b) find an equation of the tangent line to
at the given point.

a)
b)

a)
b)

a)
b)

a)
b)

a)
b)

a)
/ b)


Higher Order Derivatives

base case: Consider the zeroth-order polynomial, .
induction step: Suppose that the n-th derivative of a (n-1)th order polynomial is 0. Consider the n-th order polynomial, . We can write
where
is a (n-1)th polynomial.
L'Hôpital's Rule Solutions
L'Hôpital's rule Solutions





Related Rates Solutions


Known:
Take the time derivative:
Solve for :
Plug in known values:




Known:
Take the time derivative:
Solve for :
Plug in known values:


Let be the number of revolutions made and
be the distance the boat has moved toward the dock.
Known:
(each revolution adds one circumferance of distance to s)
Solve for :
Take the time derivative:
Plug in known values:


Applications of Derivatives Cumulative Exercise Set Solutions
Relative Extrema
Find the relative maximum(s) and minimum(s), if any, of the following functions.

There are no roots of the derivative. The derivative fails to exist when x=-1 , but the function also fails to exists at that point, so it is not an extremum. Thus, the function has no relative extrema.

There are no roots of the derivative. The derivative fails to exist at .
. The point
is a minimum since
is nonnegative because of the even numerator in the exponent. The function has no relative maximum.

Since the second derivative is positive, corresponds to a relative minimum.
The derivative fails to exist when , but so does the function. There is no relative maximum.

Since the second derivative of at
is positive,
corresponds to a relative mimimum.
Since the second derivative of at
is negative,
corresponds to a relative maximum.

Since the second derivative is positive, corresponds to a relative minimum. There is no relative maximum.

Since is positive,
corresponds to a relative minimum.
Since is negative,
corresponds to a relative maximum.
Range of Function

Since is negative,
corresponds to a relative maximum.
For ,
is positive, which means that the function is increasing. Coming from very negative
-values,
increases from a very negative value to reach a relative maximum of
at
.
For ,
is negative, which means that the function is decreasing.
Since is positive,
corresponds to a relative minimum.
Between the function decreases from
to
, then jumps to
and decreases until it reaches a relative minimum of
at
.
For ,
is positive, so the function increases from a minimum of
.
The above analysis shows that there is a gap in the function's range between and
.
Absolute Extrema
Determine the absolute maximum and minimum of the following functions on the given domain

![[0,3]](../../../upload.wikimedia.org/math/e/d/9/ed9c05fe24c0f49f5d73f494a921e0c4.png)
is differentiable on
, so the extreme value theorem guarantees the existence of an absolute maximum and minimum on
. Find and check the critical points:
Check the endpoint:
Maximum at ; minimum at

![[-\frac{1}{2},2]](../../../upload.wikimedia.org/math/6/c/1/6c1151423823bf7287ccb7f52b600ce3.png)
Maximum at ; minimum at
Determine Intervals of Change
Find the intervals where the following functions are increasing or decreasing

is the equation of a line with negative slope, so
is positive for
and negative for
.
This means that the function is increasing on and decreasing on
.

is the equation of a bowl-shaped parabola that crosses the
-axis at
and
, so
is negative for
and positive elsewhere.
This means that the function is decreasing on and increasing elsewhere.

is the equation of a hill-shaped parabola that crosses the
-axis at
and
, so
is positive for
and negative elsewhere.
This means that the function is increasing on and decreasing elsewhere.

If you did the previous exercise then no calculation is required since this function has the same derivative as that function and thus is increasing and decreasing on the same intervals; i.e., the function is increasing on and decreasing elsewhere.

is negative on
and positive elsewhere.
So is decreasing on
and increasing elsewhere.

is decreasing on
and increasing elsewhere.
Determine Intervals of Concavity
Find the intervals where the following functions are concave up or concave down

The function is concave down everywhere.

When ,
is negative, and when
,
is positive.
This means that the function is concave down on and concave up on
.

is positive when
and negative when
.
This means that the function is concave up on and concave down on
.

If you did the previous exercise then no calculation is required since this function has the same second derivative as that function and thus is concave up and concave down on the same intervals; i.e., the function is concave up on and concave down on
.

is positive when
and negative when
.
This means that the function is concave down on and concave up on
.

is positive when
and negative when
.
This means that the function is concave down on and concave up on
.
Word Problems


Velocity is the rate in change of position with respect to time. The raptor's velocity relative to you is given by
After 4 seconds, the rate of change in position with respect to time is
Set up a coordinate system with the origin at the intersection and the -axis pointing north. We assume that the position of the bike heading north is a function of the position of the bike heading east.
The distance between the bikes is given by
Let represent the elapsed time in hours. We want
when
. Apply the chain rule to
:
Thus, the bikes are moving away from one another at 13 mph.



The volume of the can as a function of the radius, , and the height,
, is
We are constricted to have a can with a volume of , so we use this fact to relate the radius and the height:
The surface area of the side is
and the cost of the side is
The surface area of the top and bottom (which is also the cost) is
The total cost is given by
We want to minimize , so take the derivative:
Find the critical points:
Check the second derivative to see if this point corresponds to a maximum or minimum:
Since the second derivative is positive, the critical point corresponds to a minimum. Thus, the minimum cost is
Integration Solutions
Indefinite Integral Solutions

We need to find a function, , such that
We know that
So we need to find a constant, , such that
Solving for , we get
So
Check your answer by taking the derivative of the function you've found and checking that it matches the integrand:

We know that
We need to find a constant, , such that
Solving for , we get
So the general antiderivative will be
Check your answer by taking the derivative of the antiderivative you've found and checking that you get back the function you started with:




Since ,
and

Let ,
so that



;

Let ;
Then and
To evaluate , make the substitution
;
;
. Then
. So
Definite Integral Solutions



Lower bound:
Upper bound:



Lower bound:
Upper bound:




From the earlier examples we know that and that
. From this we can deduce

In exercise 1 we found that
and in exercise 2 we found that
From this we can deduce that


From the property of linearity of the endpoints we have
Make the substitution .
when
and
when
. Then
where the last step has used the evenness of . Since
is just a dummy variable, we can replace it with
. Then
Integration Cumulative Exercise Set Solutions
Integration of Polynomials
Evaluate the following:





Indefinite Integration
Find the general antiderivative of the following:








Let
Then
Integration by parts





(a)
(b)
Notice that the answers in parts (a) and (b) are not equal. However, since indefinite integrals include a constant term, we expect that the answers we found will differ by a constant. Indeed
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Sequences and Series Solutions
Calculus/Sequences and Series/Solutions </noinclude> <noinclude>
Multivariable and Differential Calculus Solutions
Calculus/Multivariable and differential calculus/Solutions </noinclude>
References
Table of Trigonometry
Definitions
Pythagorean Identities
Double Angle Identities
Angle Sum Identities
Product-to-sum identities
See also
Summation notation
Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum of a set of numbers.
- Example
Let f be a function and N,M are integers with
. Then
We say N is the lower limit and M is the upper limit of the sum.
We can replace the letter i with any other variable. For this reason i is referred to as a dummy variable. So
Conventionally we use the letters i, j, k, m for dummy variables.
- Example
Here, the dummy variable is i, the lower limit of summation is 1, and the upper limit is 5.
- Example
Sometimes, you will see summation signs with no dummy variable specified, e.g.,
In such cases the correct dummy variable should be clear from the context.
You may also see cases where the limits are unspecified. Here too, they must be deduced from the context.
Common summations
Tables of Integrals
Rules
Powers
Trigonometric Functions
Basic Trigonometric Functions
Reciprocal Trigonometric Functions
Inverse Trigonometric Functions
Exponential and Logarithmic Functions
Inverse Trigonometric Functions
Further Resources
Tables of Derivatives
General Rules
Powers and Polynomials
Trigonometric Functions
Exponential and Logarithmic Functions
Inverse Trigonometric Functions
Hyperbolic and Inverse Hyperbolic Functions
Acknowledgements and Further Reading
Acknowledgements
Portions of this book have been copied from relevant Wikipedia articles.
Contributors
In alphabetical order (by surname or display name):
- Aaron Paul (AKA Grimm)
- "Professor M." (no user page available)
- Chaotic llama
- User:Cronholm144
- User:Fephisto
- User:Juliusross
- User:Stranger104
- User:Whiteknight
Further Reading
The following books list Calculus as a prerequisite:
Other Calculus Textbooks
Other calculus textbooks available online:
- Calculus Refresher by Paul Garrett, notes on first-year calculus (PDF/TeX).
- Difference Equations to Differential Equations: An Introduction to Calculus by Dan Sloughter, available under a Creative Commons license (PDF).
- The Calculus of Functions of Several Variables by Dan Sloughter, available under a Creative Commons license (PDF).
- Lecture Notes for Applied Calculus (PDF) by Karl Heinz Dovermann, first-semester calculus without using limits.
- Elements of the Differential and Integral Calculus by William Granville (1911), a classic calculus textbook now available online in various forms. (It is also partially available at Wikisource.)
- Calculus (3rd Ed., 1994) by Michael Spivak, is a more rigorous introductory calculus textbook.
- Top-down Calculus by S. Gill Williamson, available under a Creative Commons license (PDF).
Other printed calculus textbooks:
- Apostol, Tom M. Calculus.; This two-volume set provides a rigorous introduction to calculus.
Using infinitesimals
- Elementary Calculus: An Approach Using Infinitesimals (2nd Ed., 1986) by H. Jerome Keisler, an out-of-print nonstandard calculus textbook now available online under a Creative Commons license (PDF).
- Yet Another Calculus Text by Dan Sloughter, an introduction to calculus using infinitesimals available under a Creative Commons license (PDF).
- Calculus by Benjamin Crowell, an introduction to calculus available under a Creative Commons license (PDF). Also see Crowell's "Five Free Calculus Textbooks" (2004) review on Slashdot.
Interactive Websites
- Online interactive exercises on derivatives
- Visual Calculus - Interactive Tutorial on Derivatives, Differentiation, and Integration
- Planet Math: A wiki-style math reference.
Other Resources
- Notes on Integration Techniques: Structured, comprehensive lecture notes scanned and uploaded to Wikimedia Commons.
References
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