 # The Calculus Lifesaver: All the Tools You Need to Excel at Calculus

Edition: SCH - School edition
Pages: 752
https://www.jstor.org/stable/j.ctt7s1h6

1. Front Matter
(pp. i-vi)
(pp. vii-xvii)
3. WELCOME!
(pp. xviii-xxii)
4. ACKNOWLEDGMENTS
(pp. xxiii-xxiv)
5. CHAPTER 1 Functions, Graphs, and Lines
(pp. 1-24)

Trying to do calculus without using functions would be one of the most pointless things you could do. If calculus had an ingredients list, functions would be first on it, and by some margin too. So, the first two chapters of this book are designed to jog your memory about the main features of functions. This chapter contains a review of the following topics:

functions: their domain, codomain, and range, and the vertical line test;

inverse functions and the horizontal line test;

composition of functions;

odd and even functions;

graphs of linear functions and polynomials in general, as well as...

6. CHAPTER 2 Review of Trigonometry
(pp. 25-40)

To do calculus, you really need to know trigonometry. Truth be told, we won’t see much trig at first, but when it comes, it doesn’t let up. So we might as well do a thorough review of the most important aspects of trig:

angles in radians and the basics of the trig functions;

trig functions on the real line (not just angles between 0° and 90°);

graphs of trig functions; and

trig identities.

Time to refresh your memory. . . .

The first thing I want to remind you about is the notion of radians. Instead of saying that there...

7. CHAPTER 3 Introduction to Limits
(pp. 41-56)

Calculus wouldn’t exist without the concept of limits. This means that we are going to spend a lot of time looking at them. It turns out that it’s pretty tricky to define a limit properly, but you can get an intuitive understanding of limits even without going into the gory details. This will be enough to tackle differentiation and integration. So, this chapter contains only the intuitive version; check out Appendix A for the formal version. All in all, here’s what we’ll look at in this chapter:

an intuitive idea of what a limit is;

left-hand, right-hand, and two-sided limits,...

8. CHAPTER 4 How to Solve Limit Problems Involving Polynomials
(pp. 57-74)

In the previous chapter, we looked at limits from a mostly conceptual viewpoint. Now it’s time to see some of the techniques used to evaluate limits. For the moment, we’ll concentrate on limits involving polynomials; later on we’ll see how to deal with trig functions, exponentials, and logarithms. As we’ll see in the next chapter, differentiation involves taking limits of ratios, so most of our focus will be on this type of limit.

When you’re taking the limit of a ratio of two polynomials, it’s really important to notice where the limit is being taken. In particular, the techniques for...

9. CHAPTER 5 Continuity and Differentiability
(pp. 75-98)

In general, there’s only one special thing about the graph of a function: it just has to obey the vertical line test. That’s not particularly exclusive. The graph could be all over the place—a little bit here, a vertical asymptote there, or any number of individual disconnected points wherever the hell they feel like being. So now we’re going to see what happens if we’re a little more exclusive: we want to look at two types ofsmoothness. First, continuity: intuitively, this means that the graph now has to be drawn in one piece, without taking the pen off...

10. CHAPTER 6 How to Solve Differentiation Problems
(pp. 99-126)

Now we’ll see how to apply some of the theory from the previous chapter to solve problems involving differentiation. Finding derivatives from the formula is possible but cumbersome, so we’ll look at a few rules that make life a lot easier. All in all, here’s what we’ll tackle in this chapter:

finding derivatives using the definition;

using the product, quotient, and chain rules;

finding equations of tangent lines;

velocity and acceleration;

finding limits which are derivatives in disguise;

how to differentiate piecewise-defined functions; and

using the graph of a function to draw the graph of its derivative.

Let’s say we...

11. CHAPTER 7 Trig Limits and Derivatives
(pp. 127-148)

So far, most of our limits and derivatives have involved only polynomials or poly-type functions. Now let’s expand our horizons by looking at trig functions. In particular, we’ll focus on the following topics:

the behavior of trig functions at small, large, and other argument values;

derivatives of trig functions; and

simple harmonic motion.

Consider the following two limits:

$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin (5x)}{x}\quad \quad \text{and}\quad \quad \underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin (5x)}{x}.$

They look almost the same. The only difference is that the first limit is taken asx→ 0 while the second is taken asx→ ∞. What a difference, though! As we’ll soon see, the answers and the techniques...

12. CHAPTER 8 Implicit Differentiation and Related Rates
(pp. 149-166)

Let’s take a break from trying to work out how to differentiate everything in sight. It’s time to look at implicit differentiation, which is a nice generalization of regular differentiation. We’ll then see how to use this technique to solve word problems involving changing quantities. Knowing how fast one quantity is changing allows us to find how fast a different, but related, quantity is changing too. Anyway, the summary for this chapter is the same as the title:

implicit differentiation; and

related rates.

Consider the following two derivatives:

$\frac{d}{dx}\left( {{x}^{2}} \right)\quad \quad \text{and}\quad \frac{d}{dx}\left( {{y}^{2}} \right)$

The first is just 2x, as we’ve seen. So isn’t the...

13. CHAPTER 9 Exponentials and Logarithms
(pp. 167-200)

Here’s a big old chapter on exponentials and logarithms. After we review the properties of these functions, we need to do some calculus with them. It turns out that there’s a special base, the numbere, that works out particularly nicely. In particular, doing calculus withexand loge(x) is a little easier than dealing with 2xand log3(x), for example. So we need to spend some time looking ate. There are other things we want to look at as well; all in all, the plan is to check out the following topics:

review of the basics of exponentials...

14. CHAPTER 10 Inverse Functions and Inverse Trig Functions
(pp. 201-224)

In the previous chapter, we looked at exponentials and logarithms. We got a lot of mileage out of the fact thatexand ln(x) are inverses of each other. In this chapter, we’ll look at some more general properties of inverse functions, then examine inverse trig functions (and their hyperbolic cousins) in greater detail. Here’s the game plan:

using the derivative to show that a function has an inverse;

finding the derivative of inverse functions;

inverse trig functions, one by one; and

inverse hyperbolic functions.

In Section 1.2 of Chapter 1, we reviewed the basics of inverse functions. I strongly...

15. CHAPTER 11 The Derivative and Graphs
(pp. 225-244)

We have seen how to differentiate functions from several different families: polynomials and poly-type functions, trig and inverse trig functions, exponentials and logs, and even hyperbolic functions and their inverses. Now we can use this knowledge to help us sketch graphs of functions in general. We’ll see how the derivative helps us understand the maxima and minima of functions, and how the second derivative helps us to understand the so-called concavity of functions. All in all, we have the following agenda:

global and local maxima and minima (that is, extrema) of functions, and how to find them using the derivative;...

16. CHAPTER 12 Sketching Graphs
(pp. 245-266)

Now it’s time to look at a general method for sketching the graph ofy=f(x) for some given functionf. When we sketch a graph, we’re not looking for perfection; we just want to illustrate the main features of the graph. Indeed, we’re going to use the calculus tools we’ve developed: limits to understand the asymptotes, the first derivative to understand maxima and minima, and the second derivative to investigate the concavity. Here’s what we’ll look at:

the useful technique of making a table of signs;

a general method for sketching graphs; and

five examples of how to...

17. CHAPTER 13 Optimization and Linearization
(pp. 267-292)

We’re now going to look at two practical applications of calculus: optimization and linearization. Believe it or not, these techniques are used every day by engineers, economists, and doctors, for example. Basically, optimization involves finding the best situation possible, whether that be the cheapest way to build a bridge without it falling down or something as mundane as finding the fastest driving route to a specific destination. On the other hand, linearization is a useful technique for finding approximate values of hard-to-calculate quantities. It can also be used to find approximate values of zeroes of functions; this is called Newton’s...

18. CHAPTER 14 LʹHôpitalʹs Rule and Overview of Limits
(pp. 293-306)

We’ve used limits to find derivatives. Now we’ll turn things upside-down and use derivatives to find limits, by way of a nice technique called l’Hôpital’s Rule. After looking at various varieties of the rule, we’ll give a summary, followed by an overview of all the methods we’ve used so far to evaluate limits. So, we’ll look at:

l’Hôpital’s Rule, and four types of limits which naturally lead to using the rule; and

a summary of limit techniques from earlier chapters.

Most of the limits we’ve looked at are naturally in one of the following forms:

$\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)},\quad \quad \underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)-g\left( x \right) \right),\quad \quad \underset{x\to a}{\mathop{\lim }}\,f\left( x \right)g\left( x \right),\quad \quad \text{and}\quad \quad \underset{x\to a}{\mathop{\lim }}\,f{{\left( x \right)}^{g\left( x \right)}}$.

Sometimes you can just...

19. CHAPTER 15 Introduction to Integration
(pp. 307-324)

So far as calculus is concerned, differentiation is only half the story. The other half concerns integration. This powerful tool enables us to find areas of curved regions, volumes of solids, and distances traveled by objects moving at variable speeds. In this chapter, we’ll spend some time developing the theory we need to define the definite integral. Then, in the next chapter, we’ll give the definition and see how to apply it. So here’s the plan for the preliminaries on integration:

sigma notation and telescoping sums;

the relationship between displacement and area; and

using partitions to find areas.

Consider the...

20. CHAPTER 16 Definite Integrals
(pp. 325-354)

Now it’s time to get some facts straight about definite integrals. First we’ll give an informal definition in terms of areas; then we’ll use our ideas about partitions from the previous chapter to tighten up the definition. After one (exhausting) example of applying the tightened-up definition, we’ll see what else we can say about definite integrals. More precisely, we’ll look at the following topics:

signed areas and definite integrals;

the definition of the definite integral;

an example using this definition;

basic properties of definite integrals;

using integrals to find unsigned areas, the area between two curves, and areas between a...

21. CHAPTER 17 The Fundamental Theorems of Calculus
(pp. 355-382)

Here it is: the big kahuna. I’m talking about the Fundamental Theorems of Calculus, which not only provide the key for finding definite integrals without using messy Riemann sums, but also show how differentiation and integration are connected to each other. Without further ado, here’s the roadmap for the chapter: we’ll investigate

functions which are based on integrals of other functions;

the First Fundamental Theorem, and the basic idea of antiderivatives;

the Second Fundamental Theorem; and

indefinite integrals and their properties.

After all this theoretical stuff, we’ll look at a lot of different examples in the following categories:

problems based...

22. CHAPTER 18 Techniques of Integration, Part One
(pp. 383-408)

Let’s kick off the process of building up a virtual toolkit of techniques to find antiderivatives. In this chapter, we’ll look at the following three techniques:

the method of substitution (otherwise known as “change of variables”);

integration by parts; and

using partial fractions to integrate rational functions.

Then, in the next chapter, we’ll look at some more techniques involving trig functions.

Using the chain rule, we can easily differentiateex2with respect toxand see that

$\frac{d}{dx}\left( {{e}^{{{x}^{2}}}} \right)\ =\ 2x{{e}^{{{x}^{2}}}}$.

The factor 2xis the derivative ofx2, which appears in the exponent. Now, as we saw in Section 17.4 of the...

23. CHAPTER 19 Techniques of Integration, Part Two
(pp. 409-430)

In this chapter, we’ll finish gathering our techniques of integration by taking an extensive look at integrals involving trig functions. Sometimes one has to use trig identities to solve these types of problems; on other occasions there are no trig functions present, so you have to introduce some by making a trig substitution. After we finish all this trigonometry, there’ll be a quick wrap-up of the techniques from this and the previous chapter so that you can keep it all together. So, this is what we’ll look at in this chapter:

integrals involving trig identities;

integrals involving powers of trig...

24. CHAPTER 20 Improper Integrals: Basic Concepts
(pp. 431-450)

This is a difficult topic, so I’m devoting two chapters to it. This chapter serves as an introduction to improper integrals. The next chapter gets into the details of how to solve problems involving improper integrals. If you are reading this chapter for the first time, you should probably take care to try to understand all the points in it. On the other hand, if you are reviewing for a test, most likely you’ll want to skim over the chapter, noting the boxed formulas and the sections marked as important, and concentrate on the next chapter. Here’s what we’ll actually...

25. CHAPTER 21 Improper Integrals: How to Solve Problems
(pp. 451-476)

Let’s get practical and look at a lot of examples of improper integrals. As we go along, we’ll summarize the main methods. In the previous chapter, we introduced some tests that will turn out to be really useful. To use them effectively, you have to understand how some common functions behave, especially near 0 and near ∞. By “common functions,” I mean our usual suspects: polynomials, trig functions, exponentials, and logarithms. So, here’s the game plan for this chapter:

what to do when you first see an improper integral, including how to deal with multiple problem spots and functions which...

26. CHAPTER 22 Sequences and Series: Basic Concepts
(pp. 477-500)

Here’s the good news: infinite series are pretty similar to improper integrals. So a lot, but not all, of the relevant techniques are shared and we don’t need to reinvent the wheel. In order to define what an infinite series is, we’ll also need to look at sequences. Just as in the case of improper integrals, I’m devoting two chapters to sequences and series: this first chapter covers general principles, while the next one is more practical and contains methods for solving problems. If you’re reading this for the first time, go ahead and check out the details of this...

27. CHAPTER 23 How to Solve Series Problems
(pp. 501-518)

The scenario: you are given a series$\sum\nolimits_{n=1}^{\infty }{\ {{a}_{n}}}$,and you want to know whether or not it converges. If it does converge, then perhaps you’d like to know its value (that is, what it converges to). The series has to be pretty special in order to find a nice expression for its value. Of course, the series may not start atn= 1 as in the above series—it could ben= 0 or some other value ofn.

This chapter is all about giving you a blueprint of how to proceed. Here’s a possible flowchart for how...

28. CHAPTER 24 Introduction to Taylor Polynomials, Taylor Series, and Power Series
(pp. 519-534)

We now come to the important topics of power series and Taylor polynomials and series. In this chapter, we’ll see a general overview of these topics. The following two chapters will deal with problem-solving techniques in the context of the material in this chapter. Here’s what we’ll look at first:

approximations, Taylor polynomials, and a Taylor approximation theorem;

how good our approximations are, and the full Taylor Theorem;

the definition of power series;

the definition of Taylor series and Maclaurin series; and

convergence issues involving Taylor series.

Here’s a nice fact: for any real numberx, we have

${{e}^{x}}\ \cong \ 1\ +\ x\ +\ \frac{{{x}^{2}}}{2}\ +\ \frac{{{x}^{3}}}{6}$.

Also,...

29. CHAPTER 25 How to Solve Estimation Problems
(pp. 535-550)

In the previous chapter, we showed how Taylor polynomials can be used to estimate (or approximate, if you prefer) certain quantities. We also saw that the remainder term could be used to get an idea of how good the approximation actually is. In this chapter, we’ll develop these techniques and look an number of examples. So, here’s the plan for the chapter:

a review of the most important facts about Taylor polynomials and series;

how to find Taylor polynomials and series;

estimation problems; and

a different method for analyzing the error.

Here are the most important facts about Taylor polynomials...

30. CHAPTER 26 Taylor and Power Series: How to Solve Problems
(pp. 551-574)

In this chapter, we’ll look at how to solve four different classes of problems involving Taylor series, Taylor polynomials and power series:

how to find where power series converge or diverge;

how to manipulate Taylor series to get other Taylor series or Taylor polynomials;

using Taylor series or Taylor polynomials to find derivatives; and

using Maclaurin series to find limits.

Let’s say we have a power series aboutx=a:

$\sum\limits_{n=0}^{\infty }{{{a}_{n}}{{\left( x\ -\ a \right)}^{n}}}$.

As we saw in the case of geometric series, a power series might converge for somexand diverge for otherx. The question that we want to...

31. CHAPTER 27 Parametric Equations and Polar Coordinates
(pp. 575-594)

So far, we’ve sketched the graphs of many equations of the formy=f(x) with respect to Cartesian coordinates. Now we’re going to look at things in a different way: first, we’ll look at what happens when the coordinatesxandyare not directly related, but are instead related by a common parameter; and then we’ll see what happens when we replace the whole darn coordinate system with something entirely different. Of course, we have to do some calculus too. So here’s the program for this chapter:

parametric equations, graphs and finding tangents;

converting from polar coordinates to...

32. CHAPTER 28 Complex Numbers
(pp. 595-616)

Why should some quadratics have all the fun? The quadraticx2− 1 gets the privilege of having two roots (1 and −1), but poor oldx2+1 doesn’t have any, since its discriminant is negative. To even things up a little, let’s introduce the concept of complex numbers. Using complex numbers, any quadratic has two roots.*(You have to count the double rootaof (xa)2as two roots.) Anyway, here’s what we’re going to be doing with complex numbers:

the complex plane, and Cartesian and polar forms...

33. CHAPTER 29 Volumes, Arc Lengths, and Surface Areas
(pp. 617-644)

We have used definite integrals to find areas. Now we’re going to use them to find volumes, lengths of curves, and surface areas. For volumes and surface areas, we’ll pay special attention to solids which are formed by revolving a region in the plane about some axis which lies in the plane; such solids are calledsolids of revolution. In the case of volumes, we’ll also look at some more general solids. Here, then, is the game plan for this chapter:

finding volumes of solids of revolution using the disc and shell methods;

finding volumes of more general solids;

finding...

34. CHAPTER 30 Differential Equations
(pp. 645-668)

A differential equation is an equation involving derivatives. These things are really useful for describing how quantities change in the real world. For example, if you want to understand how fast a population grows, or even how quickly you can pay off a student loan, a differential equation can help model the situation and give you a decent answer. In this final chapter, we’ll see how to solve certain types of differential equations. In particular, here’s what we’ll look at:

an introduction to differential equations;

separable first-order differential equations;

first-order linear differential equations;

first- and second-order constant-coefficient differential equations; and...

35. APPENDIX A Limits and Proofs
(pp. 669-702)
36. APPENDIX B Estimating Integrals
(pp. 703-716)
37. LIST OF SYMBOLS
(pp. 717-718)
38. INDEX
(pp. 719-728)