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About *Calculus Volume 1*

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 1 covers functions, limits, derivatives, and integration.

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Functions and Graphs

Introduction

Review of Functions

Abstract:
* Use functional notation to evaluate a function.
* Determine the domain and range of a function.
* Draw the graph of a function.
* Find the zeros of a function.
* Recognize a function from a table of values.
* Make new functions from two or more given functions.
* Describe the symmetry properties of a function.

Functions

Given two sets $A$ and $B,$ a set with elements that are ordered pairs $\left(x,y\right),$ where $x$ is an element of $A$ and $y$ is an element of $B,$ is a relation from $A$ to $B.$ A relation from $A$ to $B$ defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the *input*; the element of the second set is called the *output*. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

Representing Functions

Typically, a function is represented using one or more of the following tools:

Combining Functions

Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture $x$ items is described by the function $C\left(x\right)$ and the revenue created by the sale of $x$ items is described by the function $R\left(x\right),$ then the profit on the manufacture and sale of $x$ items is defined as $P\left(x\right)=R\left(x\right)-C\left(x\right).$ Using the difference between two functions, we created a new function.

Symmetry of Functions

The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function $f\left(x\right)={x}^{4}-2{x}^{2}-3$ shown in [link](a). If we take the part of the curve that lies to the right of the *y*-axis and flip it over the *y*-axis, it lays exactly on top of the curve to the left of the *y*-axis. In this case, we say the function has symmetry about the *y*-axis. On the other hand, consider the function $f\left(x\right)={x}^{3}-4x$ shown in [link](b). If we take the graph and rotate it $180\text{°}$ about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin.

Basic Classes of Functions

Abstract:
* Calculate the slope of a linear function and interpret its meaning.
* Recognize the degree of a polynomial.
* Find the roots of a quadratic polynomial.
* Describe the graphs of basic odd and even polynomial functions.
* Identify a rational function.
* Describe the graphs of power and root functions.
* Explain the difference between algebraic and transcendental functions.
* Graph a piecewise-defined function.
* Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position.

Linear Functions and Slope

The easiest type of function to consider is a linear function. Linear functions have the form $f\left(x\right)=ax+b,$ where $a$ and $b$ are constants. In [link], we see examples of linear functions when $a$ is positive, negative, and zero. Note that if $a>0,$ the graph of the line rises as $x$ increases. In other words, $f\left(x\right)=ax+b$ is increasing on $\text{(−∞, ∞)}.$ If $a<0,$ the graph of the line falls as $x$ increases. In this case, $f\left(x\right)=ax+b$ is decreasing on $\text{(−∞, ∞)}.$ If $a=0,$ the line is horizontal.

Polynomials

A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form

Algebraic Functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Transcendental Functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A *trigonometric function* relates the ratios of two sides of a right triangle. They are $\mathrm{sin}x,\mathrm{cos}x,\mathrm{tan}x,\mathrm{cot}x,\mathrm{sec}x,\text{and}{\rule{0.2em}{0ex}}\mathrm{csc}x.$ (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form $f\left(x\right)={b}^{x},$ where the base $b>0,b\ne 1.$ A logarithmic function is a function of the form $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ for some constant $b>0,b\ne 1,$ where ${\mathrm{log}}_{b}\left(x\right)=y$ if and only if ${b}^{y}=x.$ (We also discuss exponential and logarithmic functions later in the chapter.)

Piecewise-Defined Functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of $x\text{:}$

Transformations of Functions

We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function $y={x}^{2}$ to get the function $f\left(x\right)={\left(x-2\right)}^{2}.$ This subtraction represents a shift of the function $y={x}^{2}$ two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes.

Trigonometric Functions

Abstract:
* Convert angle measures between degrees and radians.
* Recognize the triangular and circular definitions of the basic trigonometric functions.
* Write the basic trigonometric identities.
* Identify the graphs and periods of the trigonometric functions.
* Describe the shift of a sine or cosine graph from the equation of the function.

Radian Measure

To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle $\theta ,$ let $s$ be the length of the corresponding arc on the unit circle ([link]). We say the angle corresponding to the arc of length 1 has radian measure 1.

The Six Basic Trigonometric Functions

Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for all angles $\theta$ for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.

Graphs and Periods of the Trigonometric Functions

We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let $P=\left(x,y\right)$ be a point on the unit circle and let $\theta$ be the corresponding angle $.$ Since the angle $\theta$ and $\theta +2\pi$ correspond to the same point $P,$ the values of the trigonometric functions at $\theta$ and at $\theta +2\pi$ are the same. Consequently, the trigonometric functions are periodic functions. The period of a function $f$ is defined to be the smallest positive value $p$ such that $f\left(x+p\right)=f\left(x\right)$ for all values $x$ in the domain of $f.$ The sine, cosine, secant, and cosecant functions have a period of $2\pi .$ Since the tangent and cotangent functions repeat on an interval of length $\pi ,$ their period is $\pi$ ([link]).

Inverse Functions

Abstract:
* Determine the conditions for when a function has an inverse.
* Use the horizontal line test to recognize when a function is one-to-one.
* Find the inverse of a given function.
* Draw the graph of an inverse function.
* Evaluate inverse trigonometric functions.

Existence of an Inverse Function

We begin with an example. Given a function $f$ and an output $y=f\left(x\right),$ we are often interested in finding what value or values $x$ were mapped to $y$ by $f.$ For example, consider the function $f\left(x\right)={x}^{3}+4.$ Since any output $y={x}^{3}+4,$ we can solve this equation for $x$ to find that the input is $x=\sqrt[3]{y-4}.$ This equation defines $x$ as a function of $y.$ Denoting this function as ${f}^{-1},$ and writing $x={f}^{-1}\left(y\right)=\sqrt[3]{y-4},$ we see that for any $x$ in the domain of $f,{f}^{-1}\left(f\left(x\right)\right)={f}^{-1}\left({x}^{3}+4\right)=x.$ Thus, this new function, ${f}^{-1},$ “undid” what the original function $f$ did. A function with this property is called the inverse function of the original function.

Finding a Function’s Inverse

We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of $f$ to elements in the range of $f.$ The inverse function maps each element from the range of $f$ back to its corresponding element from the domain of $f.$ Therefore, to find the inverse function of a one-to-one function $f,$ given any $y$ in the range of $f,$ we need to determine which $x$ in the domain of $f$ satisfies $f\left(x\right)=y.$ Since $f$ is one-to-one, there is exactly one such value $x.$ We can find that value $x$ by solving the equation $f\left(x\right)=y$ for $x.$ Doing so, we are able to write $x$ as a function of $y$ where the domain of this function is the range of $f$ and the range of this new function is the domain of $f.$ Consequently, this function is the inverse of $f,$ and we write $x={f}^{-1}\left(y\right).$ Since we typically use the variable $x$ to denote the independent variable and $y$ to denote the dependent variable, we often interchange the roles of $x$ and $y,$ and write $y={f}^{-1}\left(x\right).$ Representing the inverse function in this way is also helpful later when we graph a function $f$ and its inverse ${f}^{-1}$ on the same axes.

Inverse Trigonometric Functions

The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function ([link]). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right].$ By doing so, we define the inverse sine function on the domain $\left[-1,1\right]$ such that for any $x$ in the interval $\left[-1,1\right],$ the inverse sine function tells us which angle $\theta$ in the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ satisfies $\text{sin}{\rule{0.1em}{0ex}}\theta =x.$ Similarly, we can restrict the domains of the other trigonometric functions to define inverse trigonometric functions, which are functions that tell us which angle in a certain interval has a specified trigonometric value.

Exponential and Logarithmic Functions

Abstract:
* Identify the form of an exponential function.
* Explain the difference between the graphs of ${x}^{b}$ and ${b}^{x}.$
* Recognize the significance of the number $e.$
* Identify the form of a logarithmic function.
* Explain the relationship between exponential and logarithmic functions.
* Describe how to calculate a logarithm to a different base.
* Identify the hyperbolic functions, their graphs, and basic identities.

Exponential Functions

Exponential functions arise in many applications. One common example is population growth.

The Number *e*

A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests $P$ dollars in a savings account with an annual interest rate $r,$ compounded annually. The amount of money after 1 year is

Logarithmic Functions

Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.

Hyperbolic Functions

The hyperbolic functions are defined in terms of certain combinations of ${e}^{x}$ and ${e}^{\text{−}x}.$ These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary ([link]). If we introduce a coordinate system so that the low point of the chain lies along the $y$-axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.

Limits

Introduction

A Preview of Calculus

Abstract:
* Describe the tangent problem and how it led to the idea of a derivative.
* Explain how the idea of a limit is involved in solving the tangent problem.
* Recognize a tangent to a curve at a point as the limit of secant lines.
* Identify instantaneous velocity as the limit of average velocity over a small time interval.
* Describe the area problem and how it was solved by the integral.
* Explain how the idea of a limit is involved in solving the area problem.
* Recognize how the ideas of limit, derivative, and integral led to the studies of infinite series and multivariable calculus.

The Tangent Problem and Differential Calculus

Rate of change is one of the most critical concepts in calculus. We begin our investigation of rates of change by looking at the graphs of the three lines $f\left(x\right)=-2x-3,g\left(x\right)=\frac{1}{2}x+1,$ and $h\left(x\right)=2,$ shown in [link].

The Area Problem and Integral Calculus

We now turn our attention to a classic question from calculus. Many quantities in physics—for example, quantities of work—may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the *x*-axis over an interval ([link])?

Other Aspects of Calculus

So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in [link]. These are only a few of the types of questions that can be asked and answered using multivariable calculus. Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit.

The Limit of a Function

Abstract:
* Using correct notation, describe the limit of a function.
* Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
* Use a graph to estimate the limit of a function or to identify when the limit does not exist.
* Define one-sided limits and provide examples.
* Explain the relationship between one-sided and two-sided limits.
* Using correct notation, describe an infinite limit.
* Define a vertical asymptote.

Intuitive Definition of a Limit

Let’s first take a closer look at how the function $f\left(x\right)=\left({x}^{2}-4\right)\text{/}\left(x-2\right)$ behaves around $x=2$ in [link]. As the values of *x* approach 2 from either side of 2, the values of $y=f\left(x\right)$ approach 4. Mathematically, we say that the limit of $f\left(x\right)$ as *x* approaches 2 is 4. Symbolically, we express this limit as

The Existence of a Limit

As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.

One-Sided Limits

Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function $g\left(x\right)=|x-2|\text{/}\left(x-2\right)$ introduced at the beginning of the section (see [link](b)). As we pick values of *x* close to 2, $g\left(x\right)$ does not approach a single value, so the limit as *x* approaches 2 does not exist—that is, $\underset{x\to 2}{\text{lim}}g\left(x\right)$ DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the *x*-value 2. To provide a more accurate description, we introduce the idea of a one-sided limit. For all values to the left of 2 (or *the negative side of* 2), $g\left(x\right)=-1.$ Thus, as *x* approaches 2 from the left, $g\left(x\right)$ approaches −1. Mathematically, we say that the limit as *x* approaches 2 from the left is −1. Symbolically, we express this idea as

Infinite Limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

The Limit Laws

Abstract:
* Recognize the basic limit laws.
* Use the limit laws to evaluate the limit of a function.
* Evaluate the limit of a function by factoring.
* Use the limit laws to evaluate the limit of a polynomial or rational function.
* Evaluate the limit of a function by factoring or by using conjugates.
* Evaluate the limit of a function by using the squeeze theorem.

Evaluating Limits with the Limit Laws

The first two limit laws were stated in [link] and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.

Limits of Polynomial and Rational Functions

By now you have probably noticed that, in each of the previous examples, it has been the case that $\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right).$ This is not always true, but it does hold for all polynomials for any choice of *a* and for all rational functions at all values of *a* for which the rational function is defined.

Additional Limit Evaluation Techniques

As we have seen, we may evaluate easily the limits of polynomials and limits of some (but not all) rational functions by direct substitution. However, as we saw in the introductory section on limits, it is certainly possible for $\underset{x\to a}{\text{lim}}f\left(x\right)$ to exist when $f\left(a\right)$ is undefined. The following observation allows us to evaluate many limits of this type:

The Squeeze Theorem

The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. The next theorem, called the squeeze theorem, proves very useful for establishing basic trigonometric limits. This theorem allows us to calculate limits by “squeezing” a function, with a limit at a point *a* that is unknown, between two functions having a common known limit at *a*. [link] illustrates this idea.

Continuity

Abstract:
* Explain the three conditions for continuity at a point.
* Describe three kinds of discontinuities.
* Define continuity on an interval.
* State the theorem for limits of composite functions.
* Provide an example of the intermediate value theorem.

Continuity at a Point

Before we look at a formal definition of what it means for a function to be continuous at a point, let’s consider various functions that fail to meet our intuitive notion of what it means to be continuous at a point. We then create a list of conditions that prevent such failures.

Types of Discontinuities

As we have seen in [link] and [link], discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities. Intuitively, a removable discontinuity is a discontinuity for which there is a hole in the graph, a jump discontinuity is a noninfinite discontinuity for which the sections of the function do not meet up, and an infinite discontinuity is a discontinuity located at a vertical asymptote. [link] illustrates the differences in these types of discontinuities. Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories.

Continuity over an Interval

Now that we have explored the concept of continuity at a point, we extend that idea to continuity over an interval. As we develop this idea for different types of intervals, it may be useful to keep in mind the intuitive idea that a function is continuous over an interval if we can use a pencil to trace the function between any two points in the interval without lifting the pencil from the paper. In preparation for defining continuity on an interval, we begin by looking at the definition of what it means for a function to be continuous from the right at a point and continuous from the left at a point.

Proof

We begin by demonstrating that $\text{cos}{\rule{0.1em}{0ex}}x$ is continuous at every real number. To do this, we must show that $\underset{x\to a}{\text{lim}}\text{cos}{\rule{0.1em}{0ex}}x=\text{cos}{\rule{0.1em}{0ex}}a$ for all values of *a*.

The Intermediate Value Theorem

Functions that are continuous over intervals of the form $\left[a,b\right],$ where *a* and *b* are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem.

The Precise Definition of a Limit

Abstract:
* Describe the epsilon-delta definition of a limit.
* Apply the epsilon-delta definition to find the limit of a function.
* Describe the epsilon-delta definitions of one-sided limits and infinite limits.
* Use the epsilon-delta definition to prove the limit laws.

Quantifying Closeness

Before stating the formal definition of a limit, we must introduce a few preliminary ideas. Recall that the distance between two points *a* and *b* on a number line is given by $|a-b|.$

Proving Limit Laws

We now demonstrate how to use the epsilon-delta definition of a limit to construct a rigorous proof of one of the limit laws. The triangle inequality is used at a key point of the proof, so we first review this key property of absolute value.

One-Sided and Infinite Limits

Just as we first gained an intuitive understanding of limits and then moved on to a more rigorous definition of a limit, we now revisit one-sided limits. To do this, we modify the epsilon-delta definition of a limit to give formal epsilon-delta definitions for limits from the right and left at a point. These definitions only require slight modifications from the definition of the limit. In the definition of the limit from the right, the inequality $0 replaces $0<|x-a|<\delta ,$ which ensures that we only consider values of *x* that are greater than (to the right of) *a*. Similarly, in the definition of the limit from the left, the inequality $-\delta  replaces $0<|x-a|<\delta ,$ which ensures that we only consider values of *x* that are less than (to the left of) *a*.

Derivatives

Introduction

Defining the Derivative

Abstract:
* Recognize the meaning of the tangent to a curve at a point.
* Calculate the slope of a tangent line.
* Identify the derivative as the limit of a difference quotient.
* Calculate the derivative of a given function at a point.
* Describe the velocity as a rate of change.
* Explain the difference between average velocity and instantaneous velocity.
* Estimate the derivative from a table of values.

Tangent Lines

We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point $\left(a,f\left(a\right)\right)$ to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of $x$ near $a$ and drawing a line through the points $\left(a,f\left(a\right)\right)$ and $\left(x,f\left(x\right)\right),$ as shown in [link]. The slope of this line is given by an equation in the form of a difference quotient:

The Derivative of a Function at a Point

The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the derivative. The process of finding a derivative is called differentiation.

Velocities and Rates of Change

Now that we can evaluate a derivative, we can use it in velocity applications. Recall that if $s\left(t\right)$ is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval $\left[a,t\right]$ if $t>a$ or $\left[t,a\right]$ if $t is given by the difference quotient

The Derivative as a Function

Abstract:
* Define the derivative function of a given function.
* Graph a derivative function from the graph of a given function.
* State the connection between derivatives and continuity.
* Describe three conditions for when a function does not have a derivative.
* Explain the meaning of a higher-order derivative.

Derivative Functions

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.

Graphing a Derivative

We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since ${f}^{\prime }\left(x\right)$ gives the rate of change of a function $f\left(x\right)$ (or slope of the tangent line to $f\left(x\right)\right).$

Derivatives and Continuity

Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.

Higher-Order Derivatives

The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of $y=f\left(x\right)$ can be expressed in any of the following forms:

Differentiation Rules

Abstract:
* State the constant, constant multiple, and power rules.
* Apply the sum and difference rules to combine derivatives.
* Use the product rule for finding the derivative of a product of functions.
* Use the quotient rule for finding the derivative of a quotient of functions.
* Extend the power rule to functions with negative exponents.
* Combine the differentiation rules to find the derivative of a polynomial or rational function.

The Basic Rules

The functions $f\left(x\right)=c$ and $g\left(x\right)={x}^{n}$ where $n$ is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.

The Power Rule

We have shown that

The Sum, Difference, and Constant Multiple Rules

We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant. These rules are summarized in the following theorem.

The Product Rule

Now that we have examined the basic rules, we can begin looking at some of the more advanced rules. The first one examines the derivative of the product of two functions. Although it might be tempting to assume that the derivative of the product is the product of the derivatives, similar to the sum and difference rules, the product rule does not follow this pattern. To see why we cannot use this pattern, consider the function $f\left(x\right)={x}^{2},$ whose derivative is ${f}^{\prime }\left(x\right)=2x$ and not $\frac{d}{dx}\left(x\right)·\frac{d}{dx}\left(x\right)=1·1=1.$

The Quotient Rule

Having developed and practiced the product rule, we now consider differentiating quotients of functions. As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives; rather, it is the derivative of the function in the numerator times the function in the denominator minus the derivative of the function in the denominator times the function in the numerator, all divided by the square of the function in the denominator. In order to better grasp why we cannot simply take the quotient of the derivatives, keep in mind that

Combining Differentiation Rules

As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules. A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function.

Derivatives as Rates of Change

Abstract:
* Determine a new value of a quantity from the old value and the amount of change.
* Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
* Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
* Predict the future population from the present value and the population growth rate.
* Use derivatives to calculate marginal cost and revenue in a business situation.

Amount of Change Formula

One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point together with its rate of change at the given point. If $f\left(x\right)$ is a function defined on an interval $\left[a,a+h\right],$ then the amount of change of $f\left(x\right)$ over the interval is the change in the $y$ values of the function over that interval and is given by

Motion along a Line

Another use for the derivative is to analyze motion along a line. We have described velocity as the rate of change of position. If we take the derivative of the velocity, we can find the acceleration, or the rate of change of velocity. It is also important to introduce the idea of speed, which is the magnitude of velocity. Thus, we can state the following mathematical definitions.

Population Change

In addition to analyzing velocity, speed, acceleration, and position, we can use derivatives to analyze various types of populations, including those as diverse as bacteria colonies and cities. We can use a current population, together with a growth rate, to estimate the size of a population in the future. The population growth rate is the rate of change of a population and consequently can be represented by the derivative of the size of the population.

Changes in Cost and Revenue

In addition to analyzing motion along a line and population growth, derivatives are useful in analyzing changes in cost, revenue, and profit. The concept of a marginal function is common in the fields of business and economics and implies the use of derivatives. The marginal cost is the derivative of the cost function. The marginal revenue is the derivative of the revenue function. The marginal profit is the derivative of the profit function, which is based on the cost function and the revenue function.

Derivatives of Trigonometric Functions

Abstract:
* Find the derivatives of the sine and cosine function.
* Find the derivatives of the standard trigonometric functions.
* Calculate the higher-order derivatives of the sine and cosine.

Derivatives of the Sine and Cosine Functions

We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function $f\left(x\right),$

Derivatives of Other Trigonometric Functions

Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.

Higher-Order Derivatives

The higher-order derivatives of $\text{sin}{\rule{0.1em}{0ex}}x$ and $\text{cos}{\rule{0.1em}{0ex}}x$ follow a repeating pattern. By following the pattern, we can find any higher-order derivative of $\text{sin}{\rule{0.1em}{0ex}}x$ and $\text{cos}{\rule{0.1em}{0ex}}x.$

The Chain Rule

Abstract:
* State the chain rule for the composition of two functions.
* Apply the chain rule together with the power rule.
* Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
* Recognize the chain rule for a composition of three or more functions.
* Describe the proof of the chain rule.

Deriving the Chain Rule

When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

The Chain and Power Rules Combined

We can now apply the chain rule to composite functions, but note that we often need to use it with other rules. For example, to find derivatives of functions of the form $h\left(x\right)={\left(g\left(x\right)\right)}^{n},$ we need to use the chain rule combined with the power rule. To do so, we can think of $h\left(x\right)={\left(g\left(x\right)\right)}^{n}$ as $f\left(g\left(x\right)\right)$ where $f\left(x\right)={x}^{n}.$ Then ${f}^{\prime }\left(x\right)=n{x}^{n-1}.$ Thus, ${f}^{\prime }\left(g\left(x\right)\right)=n{\left(g\left(x\right)\right)}^{n-1}.$ This leads us to the derivative of a power function using the chain rule,

Combining the Chain Rule with Other Rules

Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned. In particular, we can use it with the formulas for the derivatives of trigonometric functions or with the product rule.

Composites of Three or More Functions

We can now combine the chain rule with other rules for differentiating functions, but when we are differentiating the composition of three or more functions, we need to apply the chain rule more than once. If we look at this situation in general terms, we can generate a formula, but we do not need to remember it, as we can simply apply the chain rule multiple times.

The Chain Rule Using Leibniz’s Notation

As with other derivatives that we have seen, we can express the chain rule using Leibniz’s notation. This notation for the chain rule is used heavily in physics applications.

Derivatives of Inverse Functions

Abstract:
* Calculate the derivative of an inverse function.
* Recognize the derivatives of the standard inverse trigonometric functions.

The Derivative of an Inverse Function

We begin by considering a function and its inverse. If $f\left(x\right)$ is both invertible and differentiable, it seems reasonable that the inverse of $f\left(x\right)$ is also differentiable. [link] shows the relationship between a function $f\left(x\right)$ and its inverse ${f}^{-1}\left(x\right).$ Look at the point $\left(a,{f}^{-1}\left(a\right)\right)$ on the graph of ${f}^{-1}\left(x\right)$ having a tangent line with a slope of ${\left({f}^{-1}\right)}^{\prime }\left(a\right)=\frac{p}{q}.$ This point corresponds to a point $\left({f}^{-1}\left(a\right),a\right)$ on the graph of $f\left(x\right)$ having a tangent line with a slope of ${f}^{\prime }\left({f}^{-1}\left(a\right)\right)=\frac{q}{p}.$ Thus, if ${f}^{-1}\left(x\right)$ is differentiable at $a,$ then it must be the case that

Derivatives of Inverse Trigonometric Functions

We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function.

Implicit Differentiation

Abstract:
* Find the derivative of a complicated function by using implicit differentiation.
* Use implicit differentiation to determine the equation of a tangent line.

Implicit Differentiation

In most discussions of math, if the dependent variable $y$ is a function of the independent variable $x,$ we express *y* in terms of $x.$ If this is the case, we say that $y$ is an explicit function of $x.$ For example, when we write the equation $y={x}^{2}+1,$ we are defining *y* explicitly in terms of $x.$ On the other hand, if the relationship between the function $y$ and the variable $x$ is expressed by an equation where $y$ is not expressed entirely in terms of $x,$ we say that the equation defines *y* implicitly in terms of $x.$ For example, the equation $y-{x}^{2}=1$ defines the function $y={x}^{2}+1$ implicitly.

Finding Tangent Lines Implicitly

Now that we have seen the technique of implicit differentiation, we can apply it to the problem of finding equations of tangent lines to curves described by equations.

Derivatives of Exponential and Logarithmic Functions

Abstract:
* Find the derivative of exponential functions.
* Find the derivative of logarithmic functions.
* Use logarithmic differentiation to determine the derivative of a function.

Derivative of the Exponential Function

Just as when we found the derivatives of other functions, we can find the derivatives of exponential and logarithmic functions using formulas. As we develop these formulas, we need to make certain basic assumptions. The proofs that these assumptions hold are beyond the scope of this course.

Derivative of the Logarithmic Function

Now that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.

Logarithmic Differentiation

At this point, we can take derivatives of functions of the form $y={\left(g\left(x\right)\right)}^{n}$ for certain values of $n,$ as well as functions of the form $y={b}^{g\left(x\right)},$ where $b>0$ and $b\ne 1.$ Unfortunately, we still do not know the derivatives of functions such as $y={x}^{x}$ or $y={x}^{\pi }.$ These functions require a technique called logarithmic differentiation, which allows us to differentiate any function of the form $h\left(x\right)=g{\left(x\right)}^{f\left(x\right)}.$ It can also be used to convert a very complex differentiation problem into a simpler one, such as finding the derivative of $y=\frac{x\sqrt{2x+1}}{{e}^{x}{\text{sin}}^{3}x}.$ We outline this technique in the following problem-solving strategy.

Applications of Derivatives

Introduction

Related Rates

Abstract:
* Express changing quantities in terms of derivatives.
* Find relationships among the derivatives in a given problem.
* Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.

Setting up Related-Rates Problems

In many real-world applications, related quantities are changing with respect to time. For example, if we consider the balloon example again, we can say that the rate of change in the volume, $V,$ is related to the rate of change in the radius, $r.$ In this case, we say that $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related rates because *V* is related to *r*. Here we study several examples of related quantities that are changing with respect to time and we look at how to calculate one rate of change given another rate of change.

Examples of the Process

Let’s now implement the strategy just described to solve several related-rates problems. The first example involves a plane flying overhead. The relationship we are studying is between the speed of the plane and the rate at which the distance between the plane and a person on the ground is changing.

Linear Approximations and Differentials

Abstract:
* Describe the linear approximation to a function at a point.
* Write the linearization of a given function.
* Draw a graph that illustrates the use of differentials to approximate the change in a quantity.
* Calculate the relative error and percentage error in using a differential approximation.

Linear Approximation of a Function at a Point

Consider a function $f$ that is differentiable at a point $x=a.$ Recall that the tangent line to the graph of $f$ at $a$ is given by the equation

Differentials

We have seen that linear approximations can be used to estimate function values. They can also be used to estimate the amount a function value changes as a result of a small change in the input. To discuss this more formally, we define a related concept: differentials. Differentials provide us with a way of estimating the amount a function changes as a result of a small change in input values.

Calculating the Amount of Error

Any type of measurement is prone to a certain amount of error. In many applications, certain quantities are calculated based on measurements. For example, the area of a circle is calculated by measuring the radius of the circle. An error in the measurement of the radius leads to an error in the computed value of the area. Here we examine this type of error and study how differentials can be used to estimate the error.

Maxima and Minima

Abstract:
* Define absolute extrema.
* Define local extrema.
* Explain how to find the critical points of a function over a closed interval.
* Describe how to use critical points to locate absolute extrema over a closed interval.

Absolute Extrema

Consider the function $f\left(x\right)={x}^{2}+1$ over the interval $\left(\text{−}\infty ,\infty \right).$ As $x\to \text{±}\infty ,$ $f\left(x\right)\to \infty .$ Therefore, the function does not have a largest value. However, since ${x}^{2}+1\ge 1$ for all real numbers $x$ and ${x}^{2}+1=1$ when $x=0,$ the function has a smallest value, 1, when $x=0.$ We say that 1 is the absolute minimum of $f\left(x\right)={x}^{2}+1$ and it occurs at $x=0.$ We say that $f\left(x\right)={x}^{2}+1$ does not have an absolute maximum (see the following figure).

Local Extrema and Critical Points

Consider the function $f$ shown in [link]. The graph can be described as two mountains with a valley in the middle. The absolute maximum value of the function occurs at the higher peak, at $x=2.$ However, $x=0$ is also a point of interest. Although $f\left(0\right)$ is not the largest value of $f,$ the value $f\left(0\right)$ is larger than $f\left(x\right)$ for all $x$ near 0. We say $f$ has a local maximum at $x=0.$ Similarly, the function $f$ does not have an absolute minimum, but it does have a local minimum at $x=1$ because $f\left(1\right)$ is less than $f\left(x\right)$ for $x$ near 1.

Locating Absolute Extrema

The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in [link], one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Therefore, by [link], the point $c$ at which the local extremum occurs must be a critical point. We summarize this result in the following theorem.

The Mean Value Theorem

Abstract:
* Explain the meaning of Rolle’s theorem.
* Describe the significance of the Mean Value Theorem.
* State three important consequences of the Mean Value Theorem.

Rolle’s Theorem

Informally, Rolle’s theorem states that if the outputs of a differentiable function $f$ are equal at the endpoints of an interval, then there must be an interior point $c$ where $f\prime \left(c\right)=0.$ [link] illustrates this theorem.

The Mean Value Theorem and Its Meaning

Rolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions $f$ defined on a closed interval $\left[a,b\right]$ with $f\left(a\right)=f\left(b\right)$. The Mean Value Theorem generalizes Rolle’s theorem by considering functions that do not necessarily have equal value at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem ([link]). The Mean Value Theorem states that if $f$ is continuous over the closed interval $\left[a,b\right]$ and differentiable over the open interval $\left(a,b\right),$ then there exists a point $c\in \left(a,b\right)$ such that the tangent line to the graph of $f$ at $c$ is parallel to the secant line connecting $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right).$

Corollaries of the Mean Value Theorem

Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections.

Derivatives and the Shape of a Graph

Abstract:
* Explain how the sign of the first derivative affects the shape of a function’s graph.
* State the first derivative test for critical points.
* Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.
* Explain the concavity test for a function over an open interval.
* Explain the relationship between a function and its first and second derivatives.
* State the second derivative test for local extrema.

The First Derivative Test

Corollary $3$ of the Mean Value Theorem showed that if the derivative of a function is positive over an interval $I$ then the function is increasing over $I.$ On the other hand, if the derivative of the function is negative over an interval $I,$ then the function is decreasing over $I$ as shown in the following figure.

Concavity and Points of Inflection

We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function.

The Second Derivative Test

The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.

Limits at Infinity and Asymptotes

Abstract:
* Calculate the limit of a function as $x$ increases or decreases without bound.
* Recognize a horizontal asymptote on the graph of a function.
* Estimate the end behavior of a function as $x$ increases or decreases without bound.
* Recognize an oblique asymptote on the graph of a function.
* Analyze a function and its derivatives to draw its graph.

Limits at Infinity

We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.

End Behavior

The behavior of a function as $x\to \text{±}\infty$ is called the function’s end behavior. At each of the function’s ends, the function could exhibit one of the following types of behavior:

Guidelines for Drawing the Graph of a Function

We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.

Applied Optimization Problems

Abstract: * Set up and solve optimization problems in several applied fields.

Solving Optimization Problems over a Closed, Bounded Interval

The basic idea of the optimization problems that follow is the same. We have a particular quantity that we are interested in maximizing or minimizing. However, we also have some auxiliary condition that needs to be satisfied. For example, in [link], we are interested in maximizing the area of a rectangular garden. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like. Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter.

Solving Optimization Problems when the Interval Is Not Closed or Is Unbounded

In the previous examples, we considered functions on closed, bounded domains. Consequently, by the extreme value theorem, we were guaranteed that the functions had absolute extrema. Let’s now consider functions for which the domain is neither closed nor bounded.

L’Hôpital’s Rule

Abstract:
* Recognize when to apply L’Hôpital’s rule.
* Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
* Describe the relative growth rates of functions.

Applying L’Hôpital’s Rule

L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider

Other Indeterminate Forms

L’Hôpital’s rule is very useful for evaluating limits involving the indeterminate forms $\frac{0}{0}$ and $\infty \text{/}\infty .$ However, we can also use L’Hôpital’s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions $0·\infty ,$ $\infty -\infty ,$ ${1}^{\infty },$ ${\infty }^{0},$ and ${0}^{0}$ are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits. Next we realize why these are indeterminate forms and then understand how to use L’Hôpital’s rule in these cases. The key idea is that we must rewrite the indeterminate forms in such a way that we arrive at the indeterminate form $\frac{0}{0}$ or $\infty \text{/}\infty .$

Growth Rates of Functions

Suppose the functions $f$ and $g$ both approach infinity as $x\to \infty .$ Although the values of both functions become arbitrarily large as the values of $x$ become sufficiently large, sometimes one function is growing more quickly than the other. For example, $f\left(x\right)={x}^{2}$ and $g\left(x\right)={x}^{3}$ both approach infinity as $x\to \infty .$ However, as shown in the following table, the values of ${x}^{3}$ are growing much faster than the values of ${x}^{2}.$

Newton’s Method

Abstract:
* Describe the steps of Newton’s method.
* Explain what an iterative process means.
* Recognize when Newton’s method does not work.
* Apply iterative processes to various situations.

Describing Newton’s Method

Consider the task of finding the solutions of $f\left(x\right)=0.$ If $f$ is the first-degree polynomial $f\left(x\right)=ax+b,$ then the solution of $f\left(x\right)=0$ is given by the formula $x=-\frac{b}{a}.$ If $f$ is the second-degree polynomial $f\left(x\right)=a{x}^{2}+bx+c,$ the solutions of $f\left(x\right)=0$ can be found by using the quadratic formula. However, for polynomials of degree $3$ or more, finding roots of $f$ becomes more complicated. Although formulas exist for third- and fourth-degree polynomials, they are quite complicated. Also, if $f$ is a polynomial of degree $5$ or greater, it is known that no such formulas exist. For example, consider the function

Failures of Newton’s Method

Typically, Newton’s method is used to find roots fairly quickly. However, things can go wrong. Some reasons why Newton’s method might fail include the following:

Other Iterative Processes

As mentioned earlier, Newton’s method is a type of iterative process. We now look at an example of a different type of iterative process.

Antiderivatives

Abstract:
* Find the general antiderivative of a given function.
* Explain the terms and notation used for an indefinite integral.
* State the power rule for integrals.
* Use antidifferentiation to solve simple initial-value problems.

The Reverse of Differentiation

At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function $f,$ how can we find a function with derivative $f?$ If we can find a function $F$ with derivative $f,$ we call $F$ an antiderivative of $f.$

Indefinite Integrals

We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function $f,$ we use the notation ${f}^{\prime }\left(x\right)$ or $\frac{df}{dx}$ to denote the derivative of $f.$ Here we introduce notation for antiderivatives. If $F$ is an antiderivative of $f,$ we say that $F\left(x\right)+C$ is the most general antiderivative of $f$ and write

Initial-Value Problems

We look at techniques for integrating a large variety of functions involving products, quotients, and compositions later in the text. Here we turn to one common use for antiderivatives that arises often in many applications: solving differential equations.

Integration

Introduction

Approximating Areas

Abstract:
* Use sigma (summation) notation to calculate sums and powers of integers.
* Use the sum of rectangular areas to approximate the area under a curve.
* Use Riemann sums to approximate area.

Sigma (Summation) Notation

As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). The Greek capital letter $\Sigma ,$ sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write

Approximating Area

Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let $f\left(x\right)$ be a continuous, nonnegative function defined on the closed interval $\left[a,b\right].$ We want to approximate the area *A* bounded by $f\left(x\right)$ above, the *x*-axis below, the line $x=a$ on the left, and the line $x=b$ on the right ([link]).

Forming Riemann Sums

So far we have been using rectangles to approximate the area under a curve. The heights of these rectangles have been determined by evaluating the function at either the right or left endpoints of the subinterval $\left[{x}_{i-1},{x}_{i}\right].$ In reality, there is no reason to restrict evaluation of the function to one of these two points only. We could evaluate the function at any point ${x}_{i}^{\*}$ in the subinterval $\left[{x}_{i-1},{x}_{i}\right],$ and use $f\left({x}_{i}^{\*}\right)$ as the height of our rectangle. This gives us an estimate for the area of the form

The Definite Integral

Abstract:
* State the definition of the definite integral.
* Explain the terms integrand, limits of integration, and variable of integration.
* Explain when a function is integrable.
* Describe the relationship between the definite integral and net area.
* Use geometry and the properties of definite integrals to evaluate them.
* Calculate the average value of a function.

Definition and Notation

The definite integral generalizes the concept of the area under a curve. We lift the requirements that $f\left(x\right)$ be continuous and nonnegative, and define the definite integral as follows.

Evaluating Definite Integrals

Evaluating definite integrals this way can be quite tedious because of the complexity of the calculations. Later in this chapter we develop techniques for evaluating definite integrals *without* taking limits of Riemann sums. However, for now, we can rely on the fact that definite integrals represent the area under the curve, and we can evaluate definite integrals by using geometric formulas to calculate that area. We do this to confirm that definite integrals do, indeed, represent areas, so we can then discuss what to do in the case of a curve of a function dropping below the *x*-axis.

Area and the Definite Integral

When we defined the definite integral, we lifted the requirement that $f\left(x\right)$ be nonnegative. But how do we interpret “the area under the curve” when $f\left(x\right)$ is negative?

Properties of the Definite Integral

The properties of indefinite integrals apply to definite integrals as well. Definite integrals also have properties that relate to the limits of integration. These properties, along with the rules of integration that we examine later in this chapter, help us manipulate expressions to evaluate definite integrals.

Average Value of a Function

We often need to find the average of a set of numbers, such as an average test grade. Suppose you received the following test scores in your algebra class: 89, 90, 56, 78, 100, and 69. Your semester grade is your average of test scores and you want to know what grade to expect. We can find the average by adding all the scores and dividing by the number of scores. In this case, there are six test scores. Thus,

The Fundamental Theorem of Calculus

Abstract:
* Describe the meaning of the Mean Value Theorem for Integrals.
* State the meaning of the Fundamental Theorem of Calculus, Part 1.
* Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
* State the meaning of the Fundamental Theorem of Calculus, Part 2.
* Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
* Explain the relationship between differentiation and integration.

The Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at some point in that interval. The theorem guarantees that if $f\left(x\right)$ is continuous, a point *c* exists in an interval $\left[a,b\right]$ such that the value of the function at *c* is equal to the average value of $f\left(x\right)$ over $\left[a,b\right].$ We state this theorem mathematically with the help of the formula for the average value of a function that we presented at the end of the preceding section.

Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives

As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The theorem is comprised of two parts, the first of which, the Fundamental Theorem of Calculus, Part 1, is stated here. Part 1 establishes the relationship between differentiation and integration.

Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem

The Fundamental Theorem of Calculus, Part 2, is perhaps the most important theorem in calculus. After tireless efforts by mathematicians for approximately 500 years, new techniques emerged that provided scientists with the necessary tools to explain many phenomena. Using calculus, astronomers could finally determine distances in space and map planetary orbits. Everyday financial problems such as calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy. Engineers could calculate the bending strength of materials or the three-dimensional motion of objects. Our view of the world was forever changed with calculus.

Integration Formulas and the Net Change Theorem

Abstract:
* Apply the basic integration formulas.
* Explain the significance of the net change theorem.
* Use the net change theorem to solve applied problems.
* Apply the integrals of odd and even functions.

Basic Integration Formulas

Recall the integration formulas given in the table in Antiderivatives and the rule on properties of definite integrals. Let’s look at a few examples of how to apply these rules.

The Net Change Theorem

The net change theorem considers the integral of a *rate of change*. It says that when a quantity changes, the new value equals the initial value plus the integral of the rate of change of that quantity. The formula can be expressed in two ways. The second is more familiar; it is simply the definite integral.

Applying the Net Change Theorem

The net change theorem can be applied to the flow and consumption of fluids, as shown in [link].

Integrating Even and Odd Functions

We saw in Functions and Graphs that an even function is a function in which $f\left(\text{−}x\right)=f\left(x\right)$ for all *x* in the domain—that is, the graph of the curve is unchanged when *x* is replaced with −*x*. The graphs of even functions are symmetric about the *y*-axis. An odd function is one in which $f\left(\text{−}x\right)=\text{−}f\left(x\right)$ for all *x* in the domain, and the graph of the function is symmetric about the origin.

Substitution

Abstract:
* Use substitution to evaluate indefinite integrals.
* Use substitution to evaluate definite integrals.

Proof

Let *f*, *g*, *u*, and *F* be as specified in the theorem. Then

Substitution for Definite Integrals

Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.

Integrals Involving Exponential and Logarithmic Functions

Abstract:
* Integrate functions involving exponential functions.
* Integrate functions involving logarithmic functions.

Integrals of Exponential Functions

The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, $y={e}^{x},$ is its own derivative and its own integral.

Integrals Involving Logarithmic Functions

Integrating functions of the form $f\left(x\right)={x}^{-1}$ result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as $f\left(x\right)=\text{ln}{\rule{0.1em}{0ex}}x$ and $f\left(x\right)={\text{log}}_{a}x,$ are also included in the rule.

Integrals Resulting in Inverse Trigonometric Functions

Abstract: * Integrate functions resulting in inverse trigonometric functions

Integrals that Result in Inverse Sine Functions

Let us begin this last section of the chapter with the three formulas. Along with these formulas, we use substitution to evaluate the integrals. We prove the formula for the inverse sine integral.

Integrals Resulting in Other Inverse Trigonometric Functions

There are six inverse trigonometric functions. However, only three integration formulas are noted in the rule on integration formulas resulting in inverse trigonometric functions because the remaining three are negative versions of the ones we use. The only difference is whether the integrand is positive or negative. Rather than memorizing three more formulas, if the integrand is negative, simply factor out −1 and evaluate the integral using one of the formulas already provided. To close this section, we examine one more formula: the integral resulting in the inverse tangent function.

Applications of Integration

Introduction

Areas between Curves

Abstract:
* Determine the area of a region between two curves by integrating with respect to the independent variable.
* Find the area of a compound region.
* Determine the area of a region between two curves by integrating with respect to the dependent variable.

Area of a Region between Two Curves

Let $f\left(x\right)$ and $g\left(x\right)$ be continuous functions over an interval $\left[a,b\right]$ such that $f\left(x\right)\ge g\left(x\right)$ on $\left[a,b\right].$ We want to find the area between the graphs of the functions, as shown in the following figure.

Areas of Compound Regions

So far, we have required $f\left(x\right)\ge g\left(x\right)$ over the entire interval of interest, but what if we want to look at regions bounded by the graphs of functions that cross one another? In that case, we modify the process we just developed by using the absolute value function.

Regions Defined with Respect to *y*

In [link], we had to evaluate two separate integrals to calculate the area of the region. However, there is another approach that requires only one integral. What if we treat the curves as functions of $y,$ instead of as functions of $x?$ Review [link]. Note that the left graph, shown in red, is represented by the function $y=f\left(x\right)={x}^{2}.$ We could just as easily solve this for $x$ and represent the curve by the function $x=v\left(y\right)=\sqrt{y}.$ (Note that $x=\text{−}\sqrt{y}$ is also a valid representation of the function $y=f\left(x\right)={x}^{2}$ as a function of $y.$ However, based on the graph, it is clear we are interested in the positive square root.) Similarly, the right graph is represented by the function $y=g\left(x\right)=2-x,$ but could just as easily be represented by the function $x=u\left(y\right)=2-y.$ When the graphs are represented as functions of $y,$ we see the region is bounded on the left by the graph of one function and on the right by the graph of the other function. Therefore, if we integrate with respect to $y,$ we need to evaluate one integral only. Let’s develop a formula for this type of integration.

Determining Volumes by Slicing

Abstract:
* Determine the volume of a solid by integrating a cross-section (the slicing method).
* Find the volume of a solid of revolution using the disk method.
* Find the volume of a solid of revolution with a cavity using the washer method.

Volume and the Slicing Method

Just as area is the numerical measure of a two-dimensional region, volume is the numerical measure of a three-dimensional solid. Most of us have computed volumes of solids by using basic geometric formulas. The volume of a rectangular solid, for example, can be computed by multiplying length, width, and height: $V=lwh.$ The formulas for the volume of a sphere $\left(V=\frac{4}{3}\pi {r}^{3}\right),$ a cone $\left(V=\frac{1}{3}\pi {r}^{2}h\right),$ and a pyramid $\left(V=\frac{1}{3}Ah\right)$ have also been introduced. Although some of these formulas were derived using geometry alone, all these formulas can be obtained by using integration.

Solids of Revolution

If a region in a plane is revolved around a line in that plane, the resulting solid is called a solid of revolution, as shown in the following figure.

The Disk Method

When we use the slicing method with solids of revolution, it is often called the disk method because, for solids of revolution, the slices used to over approximate the volume of the solid are disks. To see this, consider the solid of revolution generated by revolving the region between the graph of the function $f\left(x\right)={\left(x-1\right)}^{2}+1$ and the $x\text{-axis}$ over the interval $\left[-1,3\right]$ around the $x\text{-axis}\text{.}$ The graph of the function and a representative disk are shown in [link](a) and (b). The region of revolution and the resulting solid are shown in [link](c) and (d).

The Washer Method

Some solids of revolution have cavities in the middle; they are not solid all the way to the axis of revolution. Sometimes, this is just a result of the way the region of revolution is shaped with respect to the axis of revolution. In other cases, cavities arise when the region of revolution is defined as the region between the graphs of two functions. A third way this can happen is when an axis of revolution other than the $x\text{-axis}$ or $y\text{-axis}$ is selected.

Volumes of Revolution: Cylindrical Shells

Abstract:
* Calculate the volume of a solid of revolution by using the method of cylindrical shells.
* Compare the different methods for calculating a volume of revolution.

The Method of Cylindrical Shells

Again, we are working with a solid of revolution. As before, we define a region $R,$ bounded above by the graph of a function $y=f\left(x\right),$ below by the $x\text{-axis,}$ and on the left and right by the lines $x=a$ and $x=b,$ respectively, as shown in [link](a). We then revolve this region around the *y*-axis, as shown in [link](b). Note that this is different from what we have done before. Previously, regions defined in terms of functions of $x$ were revolved around the $x\text{-axis}$ or a line parallel to it.

Which Method Should We Use?

We have studied several methods for finding the volume of a solid of revolution, but how do we know which method to use? It often comes down to a choice of which integral is easiest to evaluate. [link] describes the different approaches for solids of revolution around the $x\text{-axis}.$ It’s up to you to develop the analogous table for solids of revolution around the $y\text{-axis}.$

Arc Length of a Curve and Surface Area

Abstract:
* Determine the length of a curve, $y=f\left(x\right),$ between two points.
* Determine the length of a curve, $x=g\left(y\right),$ between two points.
* Find the surface area of a solid of revolution.

Arc Length of the Curve *y* = *f*(*x*)

In previous applications of integration, we required the function $f\left(x\right)$ to be integrable, or at most continuous. However, for calculating arc length we have a more stringent requirement for $f\left(x\right).$ Here, we require $f\left(x\right)$ to be differentiable, and furthermore we require its derivative, ${f}^{\prime }\left(x\right),$ to be continuous. Functions like this, which have continuous derivatives, are called *smooth*. (This property comes up again in later chapters.)

Arc Length of the Curve *x* = *g*(*y*)

We have just seen how to approximate the length of a curve with line segments. If we want to find the arc length of the graph of a function of $y,$ we can repeat the same process, except we partition the $y\text{-axis}$ instead of the $x\text{-axis}.$ [link] shows a representative line segment.

Area of a Surface of Revolution

The concepts we used to find the arc length of a curve can be extended to find the surface area of a surface of revolution. Surface area is the total area of the outer layer of an object. For objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces. For curved surfaces, the situation is a little more complex. Let $f\left(x\right)$ be a nonnegative smooth function over the interval $\left[a,b\right].$ We wish to find the surface area of the surface of revolution created by revolving the graph of $y=f\left(x\right)$ around the $x\text{-axis}$ as shown in the following figure.

Physical Applications

Abstract:
* Determine the mass of a one-dimensional object from its linear density function.
* Determine the mass of a two-dimensional circular object from its radial density function.
* Calculate the work done by a variable force acting along a line.
* Calculate the work done in pumping a liquid from one height to another.
* Find the hydrostatic force against a submerged vertical plate.

Mass and Density

We can use integration to develop a formula for calculating mass based on a density function. First we consider a thin rod or wire. Orient the rod so it aligns with the $x\text{-axis,}$ with the left end of the rod at $x=a$ and the right end of the rod at $x=b$ ([link]). Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object.

Work Done by a Force

We now consider work. In physics, work is related to force, which is often intuitively defined as a push or pull on an object. When a force moves an object, we say the force does work on the object. In other words, work can be thought of as the amount of energy it takes to move an object. According to physics, when we have a constant force, work can be expressed as the product of force and distance.

Work Done in Pumping

Consider the work done to pump water (or some other liquid) out of a tank. Pumping problems are a little more complicated than spring problems because many of the calculations depend on the shape and size of the tank. In addition, instead of being concerned about the work done to move a single mass, we are looking at the work done to move a volume of water, and it takes more work to move the water from the bottom of the tank than it does to move the water from the top of the tank.

Hydrostatic Force and Pressure

In this last section, we look at the force and pressure exerted on an object submerged in a liquid. In the English system, force is measured in pounds. In the metric system, it is measured in newtons. Pressure is force per unit area, so in the English system we have pounds per square foot (or, perhaps more commonly, pounds per square inch, denoted psi). In the metric system we have newtons per square meter, also called *pascals*.

Moments and Centers of Mass

Abstract:
* Find the center of mass of objects distributed along a line.
* Locate the center of mass of a thin plate.
* Use symmetry to help locate the centroid of a thin plate.
* Apply the theorem of Pappus for volume.

Center of Mass and Moments

Let’s begin by looking at the center of mass in a one-dimensional context. Consider a long, thin wire or rod of negligible mass resting on a fulcrum, as shown in [link](a). Now suppose we place objects having masses ${m}_{1}$ and ${m}_{2}$ at distances ${d}_{1}$ and ${d}_{2}$ from the fulcrum, respectively, as shown in [link](b).

Center of Mass of Thin Plates

So far we have looked at systems of point masses on a line and in a plane. Now, instead of having the mass of a system concentrated at discrete points, we want to look at systems in which the mass of the system is distributed continuously across a thin sheet of material. For our purposes, we assume the sheet is thin enough that it can be treated as if it is two-dimensional. Such a sheet is called a lamina. Next we develop techniques to find the center of mass of a lamina. In this section, we also assume the density of the lamina is constant.

The Symmetry Principle

We stated the symmetry principle earlier, when we were looking at the centroid of a rectangle. The symmetry principle can be a great help when finding centroids of regions that are symmetric. Consider the following example.

Theorem of Pappus

This section ends with a discussion of the theorem of Pappus for volume, which allows us to find the volume of particular kinds of solids by using the centroid. (There is also a theorem of Pappus for surface area, but it is much less useful than the theorem for volume.)

Integrals, Exponential Functions, and Logarithms

Abstract:
* Write the definition of the natural logarithm as an integral.
* Recognize the derivative of the natural logarithm.
* Integrate functions involving the natural logarithmic function.
* Define the number $e$ through an integral.
* Recognize the derivative and integral of the exponential function.
* Prove properties of logarithms and exponential functions using integrals.
* Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.

The Natural Logarithm as an Integral

Recall the power rule for integrals:

Properties of the Natural Logarithm

Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.

Defining the Number *e*

Now that we have the natural logarithm defined, we can use that function to define the number $e.$

The Exponential Function

We now turn our attention to the function ${e}^{x}.$ Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by $\text{exp}{\rule{0.2em}{0ex}}x.$ Then,

Properties of the Exponential Function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of $e,$ we must verify that the usual laws of exponents hold for the function ${e}^{x}.$

General Logarithmic and Exponential Functions

We close this section by looking at exponential functions and logarithms with bases other than $e.$ Exponential functions are functions of the form $f\left(x\right)={a}^{x}.$ Note that unless $a=e,$ we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function $f\left(x\right)={a}^{x}$ in terms of the exponential function ${e}^{x}.$ We then examine logarithms with bases other than $e$ as inverse functions of exponential functions.

Exponential Growth and Decay

Abstract:
* Use the exponential growth model in applications, including population growth and compound interest.
* Explain the concept of doubling time.
* Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling.
* Explain the concept of half-life.

Exponential Growth Model

Many systems exhibit exponential growth. These systems follow a model of the form $y={y}_{0}{e}^{kt},$ where ${y}_{0}$ represents the initial state of the system and $k$ is a positive constant, called the *growth constant*. Notice that in an exponential growth model, we have

Exponential Decay Model

Exponential functions can also be used to model populations that shrink (from disease, for example), or chemical compounds that break down over time. We say that such systems exhibit exponential decay, rather than exponential growth. The model is nearly the same, except there is a negative sign in the exponent. Thus, for some positive constant $k,$ we have $y={y}_{0}{e}^{\text{−}kt}.$

Calculus of the Hyperbolic Functions

Abstract:
* Apply the formulas for derivatives and integrals of the hyperbolic functions.
* Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.
* Describe the common applied conditions of a catenary curve.

Derivatives and Integrals of the Hyperbolic Functions

Recall that the hyperbolic sine and hyperbolic cosine are defined as

Calculus of Inverse Hyperbolic Functions

Looking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. Most of the necessary range restrictions can be discerned by close examination of the graphs. The domains and ranges of the inverse hyperbolic functions are summarized in the following table.

Applications

One physical application of hyperbolic functions involves hanging cables. If a cable of uniform density is suspended between two supports without any load other than its own weight, the cable forms a curve called a catenary. High-voltage power lines, chains hanging between two posts, and strands of a spider’s web all form catenaries. The following figure shows chains hanging from a row of posts.

Table of Integrals

Basic Integrals

1. $\int {u}^{n}{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1$

Trigonometric Integrals

18. $\int {\text{sin}}^{2}u{\rule{0.2em}{0ex}}du=\frac{1}{2}u-\frac{1}{4}\text{sin}{\rule{0.2em}{0ex}}2u+C$

Exponential and Logarithmic Integrals

42. $\int u{e}^{au}{\rule{0.2em}{0ex}}du=\frac{1}{{a}^{2}}\left(au-1\right){e}^{au}+C$

Hyperbolic Integrals

49. $\int \text{sinh}{\rule{0.2em}{0ex}}u{\rule{0.2em}{0ex}}du=\text{cosh}{\rule{0.2em}{0ex}}u+C$

Inverse Trigonometric Integrals

59. $\int {\text{sin}}^{-1}u{\rule{0.2em}{0ex}}du=u{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u+\sqrt{1-{u}^{2}}+C$

Integrals Involving *a*2 + *u*2, *a* > 0

68. $\int \sqrt{{a}^{2}+{u}^{2}}{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}+{u}^{2}}+\frac{{a}^{2}}{2}\text{ln}{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

Integrals Involving *u*2 − *a*2, *a* > 0

77. $\int \sqrt{{u}^{2}-{a}^{2}}{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\text{ln}{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

Integrals Involving *a*2 − *u*2, *a* > 0

85. $\int \sqrt{{a}^{2}-{u}^{2}}{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C$

Integrals Involving 2*au* − *u*2, *a* > 0

94. $\int \sqrt{2au-{u}^{2}}{\rule{0.2em}{0ex}}du=\frac{u-a}{2}\sqrt{2au-{u}^{2}}+\frac{{a}^{2}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

Integrals Involving *a* + *bu*, *a* ≠ 0

98. $\int \frac{u{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{{b}^{2}}\left(a+bu-a\text{ln}{\rule{0.1em}{0ex}}|a+bu|\right)+C$

Table of Derivatives

General Formulas

1. $\frac{d}{dx}\left(c\right)=0$

Trigonometric Functions

9. $\frac{d}{dx}\left(\text{sin}{\rule{0.1em}{0ex}}x\right)=\text{cos}{\rule{0.1em}{0ex}}x$

Inverse Trigonometric Functions

15. $\frac{d}{dx}\left({\text{sin}}^{-1}x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$

Exponential and Logarithmic Functions

21. $\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$

Hyperbolic Functions

25. $\frac{d}{dx}\left(\text{sinh}{\rule{0.1em}{0ex}}x\right)=\text{cosh}{\rule{0.1em}{0ex}}x$

Inverse Hyperbolic Functions

31. $\frac{d}{dx}\left({\text{sinh}}^{-1}x\right)=\frac{1}{\sqrt{{x}^{2}+1}}$

Review of Pre-Calculus

Formulas from Geometry

$A=\text{area},$ $V=\text{Volume},{\rule{0.2em}{0ex}}\text{and}$ $S=\text{lateral surface area}$

Formulas from Algebra

Formulas from Trigonometry