Calculus II

339 Pages·2016·14.18 MB·english

by  W. Michael Kelley

Calculus II
by Chris Monahan
A member of Penguin Random House LLC
First American Edition, 2016
Published in the United States by DK Publishing
6081 E. 82nd Street, Indianapolis, Indiana 46250
Copyright © 2016 Dorling Kindersley Limited
A Penguin Random House Company
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001-295791-DECEMBER2016
Published in the United States by Dorling Kindersley Limited.
IDIOT’S GUIDES and Design are trademarks of Penguin Random House LLC
ISBN: 9781465454409
Library of Congress Catalog Card Number: 2016941029
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Contents
Part 1: Review of Pre-Calculus and Calculus I ............................. 1
1 Pre-Calculus Topics Used in Calculus II .............................3
Trigonometry .............................................................................................................................................4
Exponents and Logarithms ................................................................................................................6
Parametric Equations ............................................................................................................................8
Polar Coordinates .................................................................................................................................10
Geometric Sequences and Series .................................................................................................14
Partial Fractions .....................................................................................................................................15
2 Limits, Derivatives, and Basic Integration .......................19
Limits ............................................................................................................................................................20
Derivatives .................................................................................................................................................23
Implicit Differentiation .....................................................................................................................28
Mean Value Theorem ........................................................................................................................29
Relative Extremes and Concavity ..............................................................................................30
Applications of the Derivative ......................................................................................................33
Related Rates ...........................................................................................................................................36
3 Definite and Indefinite Integrals ........................................41
Indefinite Integrals...............................................................................................................................42
u-Substitutions ........................................................................................................................................46
Fundamental Theorem of Calculus ..........................................................................................50
Numerical Approximation ..............................................................................................................54
Part 2: Length, Area, and Volumes ..............................................61
4 Areas and Approximations ..................................................63
Riemann Sums ........................................................................................................................................64
From Numerical Approximations to the
True Area ...............................................................................................................................................76
Average Value of a Function ..........................................................................................................80
Area Between Two Curves .............................................................................................................81
Simpson’s Rule ........................................................................................................................................86
5 Volumes and Areas of Solids of Revolutions ..................91
Volumes of Solids with Defined Cross Sections ...............................................................92
Disks and Washers ................................................................................................................................97
Cylindrical Shell Method ..............................................................................................................106
Arc Length ..............................................................................................................................................110
Surface Area ...........................................................................................................................................113
iv Calculus II
Part 3: More Definite and Indefinite Integrals ...........................117
6 More Integration Techniques .............................................119
Integration by Parts ...........................................................................................................................120
Polynomials and Transcendentals ............................................................................................123
Two Transcendentals ........................................................................................................................129
7 Integration with Trigonometric Functions ....................133
Trigonometric Substitutions ........................................................................................................134
Case I: a2+x2 ..................................................................................................................................134
Case II: x2−a2 ................................................................................................................................136
Case III: a2−x2 ..............................................................................................................................138
Integrals of the Form sinn(x) cosm(x) (When Either m or n Is Odd) ...................139
Case I: Both m and n Are Odd.....................................................................................................139
Case II: Both m and n Are Even .................................................................................................140
Case III: Either m or n Is Odd ....................................................................................................141
Integrals with Integrands of the Form tann(x) secm(x) (m Is Even) ......................142
Integrals with Integrands of the Form tann(x) secm(x)
(m Is Odd, n Is Even) .....................................................................................................................143
8 Integration with Fractions .................................................147
Completing the Square ...................................................................................................................148
Integration by Partial Fractions .................................................................................................152
Nonrepeating Linear Factors ........................................................................................................152
Repeated Linear Factors .................................................................................................................154
Irreducible Quadratic Factors ......................................................................................................155
Part 4: The Infinite Series and More ..........................................159
9 To Infinity and Beyond ........................................................161
Improper Integrals ..............................................................................................................................162
Infinite Limits of Integration ......................................................................................................162
Discontinuities in the Integrand................................................................................................165
Comparison Test for Improper Integrals .............................................................................168
10 Parametric Equations ..........................................................171
First and Second Derivatives of Parametric Curves .....................................................172
Arc Length of a Parametric Curve ..........................................................................................178
11 Polar Coordinates .................................................................181
Slope of the Tangent Line .............................................................................................................182
Length of an Arc of a Polar Curve ...........................................................................................187
Area Under a Curve ..........................................................................................................................189
Contents v
12 Introduction to Vectors ......................................................195
Scalars and Vectors ............................................................................................................................196
Displacement, Velocity, and Acceleration ..........................................................................200
13 Differential Equations ........................................................205
Separable Differential Equations ..............................................................................................206
Linear Approximations....................................................................................................................212
Euler’s Method......................................................................................................................................214
Slope Fields .............................................................................................................................................217
First Order Linear Differential Equations ..........................................................................222
14 Infinite Sequences ...............................................................227
Convergence and Divergence of Sequences .......................................................................228
Squeeze Theorem ...............................................................................................................................231
Increasing, Decreasing, and Monotonic Sequences ......................................................233
15 Infinite Series ........................................................................237
Infinite Geometric Series ..............................................................................................................238
Tests of Convergence........................................................................................................................239
Alternating Series ...............................................................................................................................244
Estimating the Sum of Alternating Series ...........................................................................247
16 Power Series ........................................................................249
Power Series ...........................................................................................................................................250
MacLaurin Series ................................................................................................................................252
Taylor Series ..........................................................................................................................................258
Error Estimates for the MacLaurin and
Taylor Series.......................................................................................................................................260
17 Calculus II Final Exam .......................................................263
Chapter 1 .................................................................................................................................................264
Chapter 2 .................................................................................................................................................264
Chapter 3 ..................................................................................................................................................265
Chapter 4 ..................................................................................................................................................266
Chapter 5 ..................................................................................................................................................266
Chapter 6 ..................................................................................................................................................267
Chapter 7 ..................................................................................................................................................267
Chapter 8 ..................................................................................................................................................268
Chapter 9 ..................................................................................................................................................268
Chapter 10 ................................................................................................................................................268
Chapter 11 ................................................................................................................................................269
Chapter 12 ...............................................................................................................................................269
Chapter 13 ................................................................................................................................................270
Chapter 14 ................................................................................................................................................270
Chapter 15................................................................................................................................................271
Chapter 16 ................................................................................................................................................271
Solutions ...................................................................................................................................................272
Chapter 1 .............................................................................................................................................272
Chapter 2 .............................................................................................................................................272
Chapter 3 .............................................................................................................................................273
Chapter 4 .............................................................................................................................................273
Chapter 5 .............................................................................................................................................274
Chapter 6 .............................................................................................................................................274
Chapter 7 .............................................................................................................................................274
Chapter 8 .............................................................................................................................................275
Chapter 9 .............................................................................................................................................275
Chapter 10 ...........................................................................................................................................275
Chapter 11 ...........................................................................................................................................275
Chapter 12 ...........................................................................................................................................276
Chapter 13 ...........................................................................................................................................276
Chapter 14 ...........................................................................................................................................277
Chapter 15 ...........................................................................................................................................277
Chapter 16 ...........................................................................................................................................277
Appendixes
A Solutions to “You’ve Got Problems” ..............................279
B Integration Practice Problems and Solutions ..............297
C Glossary ................................................................................307
Index .......................................................................................313
Introduction
If you look up the word calculus in the dictionary, one of the first definitions given is that it is
a hard object, like a kidney stone. People used to joke that it was no wonder that the subject
was so difficult to pass. This book is intended to “soften the calculus” so you are able to better
understand Calculus II.
I have tried to avoid using the technical language of mathematics whenever I could. When I abso-
lutely had to use more technical language, I have translated those terms into everyday language
you can comprehend. Also, you’ll find plenty of examples in each chapter to help you understand
the solutions to each problem. These examples include easy-to-understand explanations as well
as the requisite mathematical notations.
However, I warn you not to be a “mathematical voyeur,” one who likes to watch the math being
done but who does not do it himself. As you are working through this book, be sure to have
paper, pencil, and your graphing calculator by your side—and use them. Read the example, do
the problem yourself, then look at my solution. You might want to hide the example solutions
before you do the problem so you cannot cheat. People who have taken calculus will tell you that
the first step in the problem is calculus, but the rest of the steps needed to solve the problem are
algebra. Be careful as you work.
Not every example problem included is easy. If I took that approach to teaching Calculus II,
I would be giving you a false sense of what to expect. Making things too easy also would deny
you the satisfaction of taking on a challenge and succeeding.
Finally, I used to tell my students to have fun when I handed them their exams. Your first reac-
tion to that might be similar to their reactions—“Wise a**!” However, I hope that after reading
this book, you will come to understand—as my students did—learning Calculus II is a chance
to show yourself what you can do when you set your mind to it and have fun with this book. You
never know what you can do until you try.
How This Book Is Organized
This book is presented in four parts:
In Part 1, Review of Pre-Calculus and Calculus I, you review a few key topics that you
covered in Pre-Calculus and come into play in Calculus II. You’ll also do a quick review of
limits and derivatives from Calculus I.
In Part 2, Length, Area, and Volumes, you learn about the applications of integration to
compute one-, two-, and three-dimensional measurements.
In Part 3, More Definite and Indefinite Integrals, you study integration techniques
beyond the notion of the simple antiderivative.
viii Calculus II
In Part 4, The Infinite Series and More, you look at topics that extend the notions of the
Rectangular Coordinate System as well as topics to extend what you have learned about integra-
tion. Finally, you end with a study of topics that are very applicable to how your calculator does
math. I’ve also included a final exam as the last chapter so you can assess your understanding of
what you’ve learned.
At the very end of the book, I’ve included solutions to the You’ve Got Problems sidebars through-
out the book. Also, you’ll find an appendix that provides you more integration practice as well as
a glossary of helpful terms.
Extras
Throughout the book, you’ll see helpful sidebars that reinforce what you’re learning. Here’s what
to look for:
CRITICAL POINT
These sidebars are meant to draw your attention to key issues in calculus
and key people who have been very influential in the development of the
topic.
DEFINITION
These sidebars help you break down the terms used in calculus so you can
better understand what is presented to you.
BE AWARE
Although I warn you about common pitfalls and dangers throughout the
book, these sidebars deserve special attention. They are also meant to
draw a special light on critical errors students often make.
YOU’VE GOT PROBLEMS
After I discuss a topic, I explain how to work out a certain type of
problem and then you get to try it on your own. These problems are
very similar to those I walk you through in the chapters, but now it’s your
turn to shine. Even though all the answers appear in Appendix A, you
should only look there to check your work.
PART
1
Review of Pre-Calculus
and Calculus I
The first part of this book is intended to refresh the work you did in Calculus I and review some
topics you might have studied in your Pre-Calculus class that are applied in Calculus II.
In addition, we review trigonometry and logarithms—topics you covered in Calculus I.
I also include material on parametric equations, the polar coordinate system, and partial
fractions.
CHAPTER
1
Pre-Calculus Topics
Used in Calculus II
Pre-Calculus is usually the last class a student takes before In This Chapter
entering Calculus I. The topics taught in Pre-Calculus vary
•
Reviewing key
from state to state, school to school, and sometimes, from
trigonometric relationships
teacher to teacher. There are a few topics which often form
used in calculus
the mainstay of the course simply because they are key build-
•
ing blocks to one’s ability to answer topics in Calculus (as Euler’s number and his
opposed to “understanding” calculus). The topics included in logarithms
this chapter are the ones my students needed to review as we • The other coordinate
were doing the calculus. system for the plane
•
Understanding x does
not always have to be the
independent variable
•
Learning still more about
fractions