How to Ace the Rest of Calculus: The Streetwise Guide

How to Ace the Rest of Calculus: The Streetwise Guide

How to Ace the Rest of Calculus: The Streetwise Guide

How to Ace the Rest of Calculus: The Streetwise Guide

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Overview

The sequel to How to Ace Calculus, How to Ace the Rest of Calculus provides humorous and highly readable explanations of the key topics of second and third semester calculus-such as sequences and series, polor coordinates, and multivariable calculus-without the technical details and fine print that would be found in a formal text.

Product Details

ISBN-13: 9781627798860
Publisher: Holt, Henry & Company, Inc.
Publication date: 03/26/2024
Series: How to Ace
Sold by: Barnes & Noble
Format: eBook
Pages: 308
File size: 11 MB
Note: This product may take a few minutes to download.

About the Author

Colin Adams is Professor of Mathematics at Williams College. He is the author of The Knot Book and winner of the Mathematical Association of America Distinguished Teaching Award for 1998. Joel Hass is Professor of Mathematics at the University of California at Davis, and Abigail Thompson is also Professor of Mathematics at the University of California at Davis. Adams, Hass, and Thompson are co-authors of How to Ace Calculus.

Colin Adams is the Thomas T. Read Professor of Mathematics at Williams College, where he has taught since 1985. He has produced a number of books that make mathematics more accessible and relatable, including How to Ace Calculus and its sequel, How to Ace the Rest of Calculus; Riot at the Calc Exam and other Mathematically Bent Stories; and Zombies&Calculus. Colin co-wrote and appears in the videos "The Great Pi vs. E Debate" and "Derivative vs. Integral: the Final Smackdown."

Adams received his undergraduate degree from MIT and his Ph.D. from the University of Wisconsin. He had held various grants for research in the area of knot theory and low-dimensional topology and has published numerous research articles. He received the Haimo National Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, and the Robert Foster Cherry Teaching Award in 2003. Adams also served as MAA Polya Lecturer (1998-2000), and as Sigma Xi Distinguished Lecturer (2000-2002).


Abigail Thompson is a Professor of Mathematics at the University of California at Davis. She has held fellowships from the Sloan Foundation and the National Science Foundation.


Joel Hass is Professor of Mathematics at the University of California at Davis. He has held fellowships from the Sloan Foundation and the National Science Foundation.

Read an Excerpt

How to Ace the Rest of Calculus: The Streetwise Guide


By Colin Adams, Joel Hass, Abigail Thompson

Henry Holt and Company

Copyright © 2001 W. H. Freeman and Company
All rights reserved.
ISBN: 978-1-62779-886-0



CHAPTER 1

Introduction


This is the second book in the series How to Ace_____: The Streetwise Guide. This volume is for students who have already taken at least one semester of calculus. It contains everything you are likely to need to know about the rest of calculus, including, among other things, multivariable calculus and sequences and series. Why did we write it? Because of the mountain of letters of support for the first book in the series, How to Ace Calculus: The Streetwise Guide, as illustrated by the following unsolicited letter:

Dear Math Types,

Who would have thought that three pencil-necked number heads could come up with a book that I would enjoy reading? But there you have it. Your How to Ace Calculus: The Streetwise Guide was a great read. In fact, I find that I like mathematics so much now that in order to give myself more time for it, I am going to give up being chair of the Hazing Committee at my fraternity. I am even considering giving up beer. Of course, then I would have to resign from the frat. But hey, I don't think I would really care anymore. Having had my eyes opened to the sublime beauty of mathematics, I have a newfound respect for the creative endeavors of humanity. This brings with it a respect for each and every individual in our diverse society. I no longer find myself judging people based on their outward appearance, such as the fact that their necks resemble pencils. I realize that physical characteristics do not reflect the immense creative potential that resides in every human being. So I thank you for changing my life. I will now dedicate myself to eradicating just the kind of attitude that I myself exemplified. I can only hope that in my small way, I can make a contribution to society that even approaches the tremendous contribution you three have made through your book. You are truly exceptional people, and I will be forever in your debt. Please feel free to call on me at any time if I can help you in any way, personally, financially, or otherwise.

With the warmest of regards, Billy Bob Wainger


P.S.: I can only hope and pray that you will consider writing a sequel to that first book that covers the rest of calculus, including sequences and series and multivariable calculus.

Okay, okay, so it isn't a real letter. We made it up. But what is important is the sentiment it expresses, a sentiment that was expressed in bag after bag of mail that we received regarding our first book. And we are thrilled that so many people expressed such strong feelings about our necks.

If you are looking at this book in the bookstore, then either:

1. You took beginning calculus and used our previous book and got an A. What are you waiting for? It worked once, didn't it?

2. You took beginning calculus and didn't use our previous book and got a poor grade. Clearly you should get this book to rectify your previous mistake. Can't hurt to buy the first book too.

3. You didn't buy our previous book and still got an A. Well, did you ever stop to think that maybe you got lucky? Better buy this book to stack the odds in your favor.

4. You bought the first book and didn't get an A. Did we forget to mention you have to go to class and take the exams?


We would hate to think you're standing around at the bookstore reading this, in a crowded aisle with nowhere to sit down and relax. You would probably be a lot more comfortable at home, sipping a cold soft drink as you casually peruse your new purchase.

We're not going to repeat all the advice we gave in the first book on how to pick your instructor, how to study, and all of that nonsense. (Although we hope that book is on your bookshelf next to this one.) The same rules apply.

This book is devoted to the specific topics of the rest of calculus. It won't replace your text. It can't. It doesn't weigh enough. But it will explain what is really going on in the course. So go to it, and have fun!

P.S.: Check out our Web site howtoace.com where you will find lots of stuff to help you get that A, including links to exam problems, math jokes, and additional explanations.

CHAPTER 2

Indeterminate Forms and Improper Integrals


2.1 Indeterminate forms

Knowing your limits is a very important skill. You have no business hanging by your fingertips from the roof of the Empire State Building unless you know you can do so safely. You have to have put in a lot of practice beforehand, hanging from the jungle gym during recess, then hanging from the bathroom windowsill, then the hometown watertower, the Space Needle in Seattle, and finally the Empire State Building. You must know what you are capable of, what your limits are.

Similarly, in math, you want to be able to figure out your limits. Given [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], you want to know that it equals 11. Or that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


But just as you may overestimate your finger strength and find your situation dire, so may you overestimate your ability to determine limits. For instance, what is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Not so obvious, is it? If we plug in x = 2, we get 0/0. Is 0/0 equal to 0, 1, ∞? Good question. In fact, the limit could be any of these. That's why we call a limit like this an indeterminate form.

The most notorious indeterminate form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


If we plug in x = 0, we get 0/0. But we really need to know this limit. It comes up in a variety of situations, in particular when trying to show that the derivative of sin x is cos x using the limit definition of the derivative. There is a rule that can be used to find such limits. This rule was discovered by Johann Bernoulli, one of the great mathematicians of the seventeenth century, but it is named for the Marquis of L'Hôpital, who was paying Bernoulli to teach him calculus and taking credit for Bernoulli's results. In fact, we authors are doing that with this book. We haven't written a word of this, although our names are on the cover. We're paying Bill Gates to write the whole thing. L'Hôpital is pronounced Low-pee-tall.

Anyway, here's

L'Hôpital's rule if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an indeterminate form, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


What is this rule trying to tell us? It says, "If you don't know what happens to f(x)/g(x) when you take the limit, just look at f(x)/g'(x) instead!"

Example 1 Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution First we check it's an indeterminate form by plugging x = 0 into the expression sinx/x and getting 0/0. So L'Hôpital's rule applies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Wow, was that easy! You have to love this rule. Let's try another!

Example 2 Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution This is an indeterminate form 0/0. So we can apply L'Hôpital to obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


In fact, we could actually do this without L'Hôpital. We could just do a little algebra to clean it up, like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Hey, whatever steams your clams.


Example 3 Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution Again an indeterminate form 0/0! They're everywhere. L'Hôpital says

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice we applied L'Hôpital's rule twice there. But we had to check that we still had an indeterminate form before we applied it the second time.

Common Mistake Applying L'Hôpital's rule to a form which is not indeterminate is a common mistake. This error can occur in any situation, but it's especially common when L'Hôpital's rule is being applied multiple times in a problem. Be sure to check at each stage that it still applies.

INDETERMINATE FORMS INVOLVING ∞

L'Hôpital's rule can also be applied if we have the indeterminate form ∞/∞.

Example 4 Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution If we evaluate the numerator and denominator at x = ∞, we get ∞/∞. So this is an indeterminate form. Let's apply L'Hôpital:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The limit is 0. How do we interpret that? This just means that the denominator [square root of x] grows faster than the numerator ln x as x goes to ∞.


OTHER INDETERMINATE FORMS

We have seen that if a limit is of the form 0/0 or ∞/∞, then we can apply L'Hôpital's rule. But that is the only time it applies.

Warning 0/∞ is not an indeterminate form. It is 0. And ∞/0 is not an indeterminate form. It is [+ or -]∞. In neither case does L'Hôpital's rule apply.

Sometimes other forms can be manipulated into the form 0/0 or ∞/∞. For example, suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is also an indeterminate form, since this product could be anything. But it's easy enough to turn this into or by writing:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Then it's straightforward to solve.

Example 5 Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].


Solution Letting x go to ∞, we see that this is of the form 0 · ∞. So we rewrite it as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


This is now the indeterminate form ∞/∞, so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Hey, whatever burps your baby.


2.2 Improper integrals

Well, we tried to avoid it, but now we have to introduce them. You know the type. They wear a T-shirt with a tux printed on it to the Calculus Cotillion. They eat the entire dinner with the dessert fork. They laugh loudly at involuntary auditory signs of digestive distress. That's right, we're talking about the improper integrals. If they weren't so important, they wouldn't get invited at all.

There are a couple of variations on this theme. First, there is the integral with unbounded interval of integration. That's right. Instead of integrating with limits from –2 to 4 or 3 to 7, these integrals have limits from 1 to ∞ or –∞ to 2, or even –∞ to ∞. For example, we could have the integral [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A second type of improper integral is one where the function being integrated goes to [+ or -]∞ somewhere over the interval of integration. Take a look at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which has this problem at x = 0.

But let's not swallow the entire fish whole. We'll start with the first type.


INFINITE LIMITS OF INTEGRATION

Let's look at the example [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It seems a little strange having a limit of integration that is infinite. What are we to make of it? Well, we know by now that definite integrals often represent areas under graphs of functions. So let's go with that viewpoint and see what it says in this case. It still makes sense to talk about the area under the curve 1/x2 for x going from 1 to ∞ (see Figure 2.1).

Now we know what you're thinking. That area goes on forever as we head out along the x-axis, so this integral must give ∞ for the answer. Easy! But hold on to your brain pan, because in fact, that's not what happens! To see what really happens, we need to figure out the correct interpretation of that upper limit of integration ∞. The right way to think of ∞ here is as a limit of numbers which are getting very large, or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So the official interpretation of our improper integral is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


So in this case, that area actually turned out to be 1.

We will always interpret an infinite limit improper integral this way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Sometimes an improper integral gives a finite number, as happened above. Then we say the improper integral converges. But sometimes the limit is ∞ or doesn't exist. Then we say the improper integral diverges.

Example Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution This doesn't look very different from the previous example, but whammo, when we take a limit to compute it, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


This one diverges.

We also define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any real number c that we want to use. We can use a birthdate or a lucky number, or 0. We'll always get the same answer, so we can use whatever is easiest to compute with.

Now for the second type of improper integral.


INFINITE INTEGRANDS

Given the chore of computing the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we certainly have a problem, since 1/x2/3 is undefined at x = 0. It explodes right there, one big volcano (see Figure 2.2).

On the other hand, the area under the curve doesn't look that big. High yes, but not that wide.

To make sense of this, we split the integral into two integrals around the problem point.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


Now, we will use the same idea as above. Since it doesn't make sense to talk about 1/2/3 at x = 0, we replace the 0 in the first integral by a limit.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


We take the limit as b approaches 0 from the left, since the interval of integration is all to the left of 0. Then we can compute the integral and take the limit:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hey, whatever roasts your chestnuts.

CHAPTER 3

Polar Coordinates


3.1 Introduction to polar coordinates

You know how in all those Cold War movies where we're submerged in a submarine and there's a Soviet sub just 50 feet away, so everybody has to be really quiet or else well get a torpedo right up the torpedo tube? And there's this one ensign sitting in front of the the radarscope, and on the scope there is this green radial ray that goes round and round. And all of the sailors are sweating like they just took showers in their clothes, because after all, they are packed into a sardine can and there's not enough room to bring inessentials like deodorant. And the captain whispers, "Not a sound. Nobody make a sound." And all the while, that line on the radarscope is going round and round. And each time it goes round, a big green dot corresponding to the enemy sub shows up on the screen. And every time that line goes round the scope beeps. Yes, that's right. It beeps like an alarm clock going off in the middle of the night. And you just have to wonder, sitting there in front of the TV, why can't the Russians hear the stupid scope? Do they have cotton stuffed in their ears? Are they confusing the beep of that scope with their own? The fact of the matter is that scope is loud enough to wake the dead. And as you sit in your home theater, you want to say to the captain, "You don't have to whisper. Nobody could hear you over the damn scope." You could sing God Bless America with your tonsils popping out the back of your throat, and the Russians would say, "Comrade, did you hear something?" "Comrade, I can't hear anything over that damn radarscope."

And of course, in at least 80 percent of these movies, the action is taking place under the polar ice cap. This is where submarines tend to bump into one another. So it's no big surprise that the coordinates used for the scope are called polar coordinates.

The ensign whispers to the captain in a very loud whisper, in order to be heard over the scope, "Captain, looks like a C class nuclear ranger. Appears to be carrying 37 men, 12 women, and one free-range chicken. She's 50 feet away and closing."

The scope always puts your own position at the center. Your first measurement of the other point is its distance from you, in this case 50 feet.

Then Radar Boy adds, "She's at 37 degrees," meaning that she is 50 feet out, at an angle of 37 degrees from the positive x-axis. We would say that the polar coordinates of the point are (r, θ) = (50, 37°).

Of course, unlike the navy, we use radians, because all mathematicians have agreed always to use radians. It makes our computations easier. If all you use angles for is to aim an occasional torpedo, then degrees are fine. But if you want to do hard-core mathematics with the big dogs, then radians are the way to go.

To describe a point in the plane in polar coordinates, we give its distance from the origin, called r, and its angle in radians going counterclockwise from the positive x-axis, which we call θ. So, instead of using (x, y), we describe a point by (r, θ), as in Figure 3.1.

In polar coordinates, a point can have more than one description. The point (1, 0) in rectangular coordinates has distance 1 from the origin and has angle 0 with the positive x-axis. So in polar coordinates, it is (r, θ) = (1, 0). But it is also given by (1, 2π), (1, 4π), etc. Even wackier is the origin, which is given by (0, π), (0, π/2), (0, 7.3π), etc.

If r is negative, we go out from the origin in the opposite direction. So for instance, in Figure 3.2, (r, θ) = (–1, π/4) is the same as the point given by (1, 5π/4).

From Figure 3.3 and the Pythagorean theorem, we see the following basic relationship between rectangular and polar coordinates:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


These four equations are worth knowing, but you needn't memorize them, since they can so easily be read off the picture. Let's try graphing some polar equations.

Example 1 Graph r = 3.

Solution You're going to like this one. We want to draw all the points (r, θ) in the plane that satisfy r = 3. Since θ doesn't occur in the equation, θ can be anything at all. It's free as a bird, no constraints, no obligations. Anything goes, let 'er rip. On the other hand, r is more restricted, being forced as it is to be 3 at all times. But if r is fixed at 3, and θ can do what it wants, our ray of length 3 out of the origin can swing all the way around the full 360°. We get all the points on a circle of radius 3 around the origin. The graph appears in Figure 3.4.

The polar equation of this circle is simpler than the usual circle equation, x2 + y2 = 9. That's one reason why people are fond of polar equations. They make it incredibly easy to describe certain common graphs, particularly circles centered at the origin.


(Continues...)

Excerpted from How to Ace the Rest of Calculus: The Streetwise Guide by Colin Adams, Joel Hass, Abigail Thompson. Copyright © 2001 W. H. Freeman and Company. Excerpted by permission of Henry Holt and Company.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

Introduction

Indeterminate Forms and Improper Integrals
2.1 Indeterminate forms
2.2 Improper integrals

Polar Coordinates
3.1 Introduction to polar coordinates
3.2 Area in polar coordinates

Infinite Series
4.1 Sequences
4.2 Limits of sequences
4.3 Series: The basic idea
4.4 Geometric series: The extroverts
4.5 The nth-term test
4.6 Integral test and p-series: More friends
4.7 Comparison tests
4.8 Alternating series and absolute convergence
4.9 More tests for convergence
4.10 Power series
4.11 Which test to apply when?
4.12 Taylor series
4.13 Taylor's formula with remainder
4.14 Some famous Taylor series

Vectors: From Euclid to Cupid
5.1 Vectors in the plane
5.2 Space: The final (exam) frontier
5.3 Vectors in space
5.4 The dot product
5.5 The cross product
5.6 Lines in space
5.7 Planes in space

Parametric Curves in Space: Riding the Roller Coaster
6.1 Parametric curves
6.2 Curvature
6.3 Velocity and acceleration

Surfaces and Graphing
7.1 Curves in the plane: A retrospective
7.2 Graphs of equations in 3-D space
7.3 Surfaces of revolution
7.4 Quadric surfaces (the -oid surfaces)

Functions of Several Variables and Their Partial Derivatives
8.1 Functions of several variables
8.2 Contour curves
8.3 Limits
8.4 Continuity
8.5 Partial derivatives
8.6 Max-min problems

cf08.7 The chain rule
8.8 The gradient and directional derivatives
8.9 Lagrange multipliers
8.10 Second derivative test

Multiple Integrals
9.1 Double integrals and limits—the technical stuff
9.2 Calculating double integrals
9.3 Double integrals and volumes under a graph
9.4 Double integrals in polar coordinates
9.5 Triple integrals
9.6 Cylindrical and spherical coordinates
9.7 Mass, center of mass, and moments
9.8 Change of coordinates

Vector Fields and the Green-Stokes Gang
10.1 Vector fields
10.2 Getting acquainted with div and curl
10.3 Line up for line integrals
10.4 Line integrals of vector fields
10.5 Conservative vector fields
10.6 Green's theorem
10.7 Integrating the divergence; the divergence theorem
10.8 Surface integrals
10.9 Stoking!

What's Going to Be on the Final?

Glossary: A Quick Guide to the Mathematical Jargon

Index

Just the Facts: A Quick Reference Guide

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