Calculus
Know-It-ALL
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Calculus
Know-It-ALL
Beginner to Advanced,
and Everything in Between
Stan Gibilisco
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To Tim, Tony, Samuel, and Bill
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About the Author
Stan Gibilisco is an electronics engineer, researcher, and mathematician. He is the author of
Algebra Know-It-ALL, a number of titles for McGraw-Hill’s Demystified series, more than 30
other technical books and dozens of magazine articles. His work has been published in several
languages.
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Contents
Preface
xv
Acknowledgment xvii
Part 1
Differentiation in One Variable
1 Single-Variable Functions 3
Mappings 3
Linear Functions 9
Nonlinear Functions 12
“Broken” Functions 14
Practice Exercises 17
2 Limits and Continuity 20
Concept of the Limit 20
Continuity at a Point 24
Continuity of a Function 29
Practice Exercises 33
3 What’s a Derivative?
35
Vanishing Increments 35
Basic Linear Functions 40
Basic Quadratic Functions 44
Basic Cubic Functions 48
Practice Exercises 52
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Contents
4 Derivatives Don’t Always Exist 55
Let’s Look at the Graph 55
When We Can Differentiate 59
When We Can’t Differentiate 63
Practice Exercises 69
5 Differentiating Polynomial Functions 71
Power Rule 71
Sum Rule 75
Summing the Powers 79
Practice Exercises 82
6 More Rules for Differentiation 84
Multiplication-by-Constant Rule 84
Product Rule 87
Reciprocal Rule 90
Quotient Rule 95
Chain Rule 99
Practice Exercises 103
7 A Few More Derivatives
Real-Power Rule
106
106
Sine and Cosine Functions 108
Natural Exponential Function 114
Natural Logarithm Function 118
Practice Exercises 124
8 Higher Derivatives 126
Second Derivative 126
Third Derivative 130
Beyond the Third Derivative
Practice Exercises 136
133
9 Analyzing Graphs with Derivatives 138
Three Common Traits 138
Graph of a Quadratic Function 141
Graph of a Cubic Function 144
Graph of the Sine Function 147
Practice Exercises 152
10 Review Questions and Answers
154
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Contents xi
Part 2
Integration in One Variable
11 What’s an Integral? 189
Summation Notation 189
Area Defined by a Curve 191
Three Applications 198
Practice Exercises 203
12 Derivatives in Reverse
205
Concept of the Antiderivative 205
Some Simple Antiderivatives 207
Indefinite Integral 211
Definite Integral 215
Practice Exercises 218
13 Three Rules for Integration 221
Reversal Rule 221
Split-Interval Rule 224
Substitution Rule 229
Practice Exercises 232
14 Improper Integrals 234
Variable Bounds 234
Singularity in the Interval 238
Infinite Intervals 244
Practice Exercises 248
15 Integrating Polynomial Functions
Three Rules Revisited 250
Indefinite-Integral Situations 253
Definite-Integral Situations 255
Practice Exercises 260
250
16 Areas between Graphs 262
Line and Curve 262
Two Curves 267
Singular Curves 270
Practice Exercises 274
17 A Few More Integrals
277
Sine and Cosine Functions 277
Natural Exponential Function 282
Reciprocal Function 289
Practice Exercises 295
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Contents
18 How Long Is the Arc? 297
A Chorus of Chords 297
A Monomial Curve 302
A More Exotic Curve 306
Practice Exercises 309
19 Special Integration Tricks
Principle of Linearity 311
Integration by Parts 313
Partial Fractions 318
Practice Exercises 323
311
20 Review Questions and Answers
Part 3
325
Advanced Topics
21 Differentiating Inverse Functions
A General Formula 377
Derivative of the Arcsine 381
Derivative of the Arccosine 384
Practice Exercises 388
377
22 Implicit Differentiation 390
Two-Way Relations 390
Two-Way Derivatives 394
Practice Exercises 402
23 The L’Hôpital Principles
404
Expressions That Tend Toward 0/0 404
Expressions That Tend Toward ± ∞ / ± ∞ 408
Other Indeterminate Limits 411
Practice Exercises 414
24 Partial Derivatives
416
Multi-Variable Functions 416
Two Independent Variables 419
Three Independent Variables 424
Practice Exercises 426
25 Second Partial Derivatives 428
Two Variables, Second Partials 428
Two Variables, Mixed Partials 431
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Contents xiii
Three Variables, Second Partials 434
Three Variables, Mixed Partials 438
Practice Exercises 440
26 Surface-Area and Volume Integrals 442
A Cylinder 442
A Cone 444
A Sphere 448
Practice Exercises 453
27 Repeated, Double, and Iterated Integrals 455
Repeated Integrals in One Variable 455
Double Integrals in Two Variables 458
Iterated Integrals in Two Variables 462
Practice Exercises 466
28 More Volume Integrals 468
Slicing and Integrating 468
Base Bounded by Curve and x Axis 470
Base Bounded by Curve and Line 475
Base Bounded by Two Curves 481
Practice Exercises 487
29 What’s a Differential Equation? 490
Elementary First-Order ODEs 490
Elementary Second-Order ODEs 493
Practice Exercises 500
30 Review Questions and Answers 502
Final Exam 541
Appendix A Worked-Out Solutions to Exercises: Chapters 1 to 9
589
Appendix B Worked-Out Solutions to Exercises: Chapters 11 to 19 631
Appendix C Worked-Out Solutions to Exercises: Chapters 21 to 29 709
Appendix D
Answers to Final Exam Questions
Appendix E
Special Characters in Order of Appearance
Appendix F Table of Derivatives
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776
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Contents
Appendix G Table of Integrals
Suggested Additional Reading
Index
779
783
785
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Preface
If you want to improve your understanding of calculus, then this book is for you. It can supplement standard texts at the high-school senior, trade-school, and college undergraduate
levels. It can also serve as a self-teaching or home-schooling supplement. Prerequisites include
intermediate algebra, geometry, and trigonometry. It will help if you’ve had some precalculus
(sometimes called “analysis”) as well.
This book contains three major sections. Part 1 involves differentiation in one variable.
Part 2 is devoted to integration in one variable. Part 3 deals with partial differentiation and
multiple integration. You’ll also get a taste of elementary differential equations.
Chapters 1 through 9, 11 through 19, and 21 through 29 end with practice exercises. You
may (and should) refer to the text as you solve these problems. Worked-out solutions appear
in Apps. A, B, and C. Often, these solutions do not represent the only way a problem can be
figured out. Feel free to try alternatives!
Chapters 10, 20, and 30 contain question-and-answer sets that finish up Parts 1, 2, and
3, respectively. These chapters will help you review the material.
A multiple-choice final exam concludes the course. Don’t refer to the text while taking
the exam. The questions in the exam are more general (and easier) than the practice exercises
at the ends of the chapters. The exam is designed to test your grasp of the concepts, not to see
how well you can execute calculations. The correct answers are listed in App. D.
In my opinion, most textbooks place too much importance on “churning out answers,”
and often fail to explain how and why you get those answers. I wrote this book to address these
problems. I’ve tried to introduce the language gently, so you won’t get lost in a wilderness of
jargon. Many of the examples and problems are easy, some take work, and a few are designed
to make you think hard.
If you complete one chapter per week, you’ll get through this course in a school year. But
don’t hurry. When you’ve finished this book, I recommend Calculus Demystified by Steven G.
Krantz and Advanced Calculus Demystified by David Bachman for further study. If Chap. 29
of this book gets you interested in differential equations, I recommend Differential Equations
Demystified by Steven G. Krantz as a first text in that subject.
Stan Gibilisco
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Acknowledgment
I extend thanks to my nephew Tony Boutelle. He spent many hours helping me proofread
the manuscript, and he offered insights and suggestions from the viewpoint of the intended
audience.
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Calculus
Know-It-ALL
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PART
1
Differentiation in One Variable
1
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CHAPTER
1
Single-Variable Functions
Calculus is the mathematics of functions, which are relationships between sets consisting of
objects called elements. The simplest type of function is a single-variable function, where the
elements of two sets are paired off according to certain rules.
Mappings
Imagine two sets of points defined by the large rectangles in Fig. 1-1. Suppose you’re interested in the subsets shown by the hatched ovals. You want to pair off the points in the top oval
with those in the bottom oval. When you do this, you create a mapping of the elements of one
set into the elements of the other set.
Domain, range, and variables
All the points involved in the mapping of Fig. 1-1 are inside the ovals. The top oval is called
the domain. That’s the set of elements that we “go out from.” In Fig. 1-1, these elements are
a through f. The bottom oval is called the range. That’s the set of elements that we “come in
toward.” In Fig. 1-1, these elements are v through z.
In any mapping, the elements of the domain and the range can be represented by variables. A nonspecific element of the domain is called the independent variable. A nonspecific
element of the range is called the dependent variable. The mapping assigns values of the dependent (or “output”) variable to values of the independent (or “input”) variable.
Ordered pairs
In Fig. 1-1, the mapping can be defined in terms of ordered pairs, which are two-item lists
showing how the elements are assigned to each other. The set of ordered pairs defined by the
mapping in Fig. 1-1 is
{(a,v ), (b,w ), (c,v ), (c,x ), (c,z ), (d,y ), (e,z ), ( f,y )}
3
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Single-Variable Functions
Domain
c
a
e
b
d
f
z
y
x
w
v
Range
Figure 1-1 A relation defines how the elements of a set are assigned to
the elements of another set.
Within each ordered pair, an element of the domain (a value of the independent variable)
is written before the comma, and an element of the range (a value of the dependent variable) is
written after the comma. Whenever you can express a mapping as a set of ordered pairs, then
that mapping is called a relation.
Are you confused?
You won’t see spaces after the commas inside of the ordered pairs, but you’ll see spaces after the commas
separating the ordered pairs in the list that make up the set. These aren’t typographical errors! That’s the
way they should be written.
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