Exercise And Problems in CALCULUS

Exercises and Problems in Calculus

John M. Erdman
Portland State University

Version August 1, 2013

c 2010 John M. Erdman




E-mail address:

,
, Contents

Preface ix

Part 1. PRELIMINARY MATERIAL 1
Chapter 1. INEQUALITIES AND ABSOLUTE VALUES 3
1.1. Background 3
1.2. Exercises 4
1.3. Problems 5
1.4. Answers to Odd-Numbered Exercises 6
Chapter 2. LINES IN THE PLANE 7
2.1. Background 7
2.2. Exercises 8
2.3. Problems 9
2.4. Answers to Odd-Numbered Exercises 10
Chapter 3. FUNCTIONS 11
3.1. Background 11
3.2. Exercises 12
3.3. Problems 15
3.4. Answers to Odd-Numbered Exercises 17

Part 2. LIMITS AND CONTINUITY 19
Chapter 4. LIMITS 21
4.1. Background 21
4.2. Exercises 22
4.3. Problems 24
4.4. Answers to Odd-Numbered Exercises 25
Chapter 5. CONTINUITY 27
5.1. Background 27
5.2. Exercises 28
5.3. Problems 29
5.4. Answers to Odd-Numbered Exercises 30

Part 3. DIFFERENTIATION OF FUNCTIONS OF A SINGLE VARIABLE 31
Chapter 6. DEFINITION OF THE DERIVATIVE 33
6.1. Background 33
6.2. Exercises 34
6.3. Problems 36
6.4. Answers to Odd-Numbered Exercises 37
Chapter 7. TECHNIQUES OF DIFFERENTIATION 39
iii

,iv CONTENTS

7.1. Background 39
7.2. Exercises 40
7.3. Problems 45
7.4. Answers to Odd-Numbered Exercises 47

Chapter 8. THE MEAN VALUE THEOREM 49
8.1. Background 49
8.2. Exercises 50
8.3. Problems 51
8.4. Answers to Odd-Numbered Exercises 52

Chapter 9. L’HÔPITAL’S RULE 53
9.1. Background 53
9.2. Exercises 54
9.3. Problems 56
9.4. Answers to Odd-Numbered Exercises 57

Chapter 10. MONOTONICITY AND CONCAVITY 59
10.1. Background 59
10.2. Exercises 60
10.3. Problems 65
10.4. Answers to Odd-Numbered Exercises 66

Chapter 11. INVERSE FUNCTIONS 69
11.1. Background 69
11.2. Exercises 70
11.3. Problems 72
11.4. Answers to Odd-Numbered Exercises 74

Chapter 12. APPLICATIONS OF THE DERIVATIVE 75
12.1. Background 75
12.2. Exercises 76
12.3. Problems 82
12.4. Answers to Odd-Numbered Exercises 84

Part 4. INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE 87

Chapter 13. THE RIEMANN INTEGRAL 89
13.1. Background 89
13.2. Exercises 90
13.3. Problems 93
13.4. Answers to Odd-Numbered Exercises 95

Chapter 14. THE FUNDAMENTAL THEOREM OF CALCULUS 97
14.1. Background 97
14.2. Exercises 98
14.3. Problems 102
14.4. Answers to Odd-Numbered Exercises 105

Chapter 15. TECHNIQUES OF INTEGRATION 107
15.1. Background 107
15.2. Exercises 108
15.3. Problems 115
15.4. Answers to Odd-Numbered Exercises 118

, CONTENTS v

Chapter 16. APPLICATIONS OF THE INTEGRAL 121
16.1. Background 121
16.2. Exercises 122
16.3. Problems 127
16.4. Answers to Odd-Numbered Exercises 130

Part 5. SEQUENCES AND SERIES 131

Chapter 17. APPROXIMATION BY POLYNOMIALS 133
17.1. Background 133
17.2. Exercises 134
17.3. Problems 136
17.4. Answers to Odd-Numbered Exercises 137

Chapter 18. SEQUENCES OF REAL NUMBERS 139
18.1. Background 139
18.2. Exercises 140
18.3. Problems 143
18.4. Answers to Odd-Numbered Exercises 144

Chapter 19. INFINITE SERIES 145
19.1. Background 145
19.2. Exercises 146
19.3. Problems 148
19.4. Answers to Odd-Numbered Exercises 149

Chapter 20. CONVERGENCE TESTS FOR SERIES 151
20.1. Background 151
20.2. Exercises 152
20.3. Problems 155
20.4. Answers to Odd-Numbered Exercises 156

Chapter 21. POWER SERIES 157
21.1. Background 157
21.2. Exercises 158
21.3. Problems 164
21.4. Answers to Odd-Numbered Exercises 166

Part 6. SCALAR FIELDS AND VECTOR FIELDS 169

Chapter 22. VECTOR AND METRIC PROPERTIES of Rn 171
22.1. Background 171
22.2. Exercises 174
22.3. Problems 177
22.4. Answers to Odd-Numbered Exercises 179

Chapter 23. LIMITS OF SCALAR FIELDS 181
23.1. Background 181
23.2. Exercises 182
23.3. Problems 184
23.4. Answers to Odd-Numbered Exercises 185

Part 7. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES 187

,vi CONTENTS

Chapter 24. PARTIAL DERIVATIVES 189
24.1. Background 189
24.2. Exercises 190
24.3. Problems 192
24.4. Answers to Odd-Numbered Exercises 193

Chapter 25. GRADIENTS OF SCALAR FIELDS AND TANGENT PLANES 195
25.1. Background 195
25.2. Exercises 196
25.3. Problems 199
25.4. Answers to Odd-Numbered Exercises 201

Chapter 26. MATRICES AND DETERMINANTS 203
26.1. Background 203
26.2. Exercises 207
26.3. Problems 210
26.4. Answers to Odd-Numbered Exercises 213

Chapter 27. LINEAR MAPS 215
27.1. Background 215
27.2. Exercises 217
27.3. Problems 219
27.4. Answers to Odd-Numbered Exercises 221

Chapter 28. DEFINITION OF DERIVATIVE 223
28.1. Background 223
28.2. Exercises 224
28.3. Problems 226
28.4. Answers to Odd-Numbered Exercises 227

Chapter 29. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLLES 229
29.1. Background 229
29.2. Exercises 232
29.3. Problems 234
29.4. Answers to Odd-Numbered Exercises 237

Chapter 30. MORE APPLICATIONS OF THE DERIVATIVE 239
30.1. Background 239
30.2. Exercises 241
30.3. Problems 243
30.4. Answers to Odd-Numbered Exercises 244

Part 8. PARAMETRIZED CURVES 245

Chapter 31. PARAMETRIZED CURVES 247
31.1. Background 247
31.2. Exercises 248
31.3. Problems 255
31.4. Answers to Odd-Numbered Exercises 256

Chapter 32. ACCELERATION AND CURVATURE 259
32.1. Background 259
32.2. Exercises 260
32.3. Problems 263

, CONTENTS vii

32.4. Answers to Odd-Numbered Exercises 265

Part 9. MULTIPLE INTEGRALS 267
Chapter 33. DOUBLE INTEGRALS 269
33.1. Background 269
33.2. Exercises 270
33.3. Problems 274
33.4. Answers to Odd-Numbered Exercises 275
Chapter 34. SURFACES 277
34.1. Background 277
34.2. Exercises 278
34.3. Problems 280
34.4. Answers to Odd-Numbered Exercises 281
Chapter 35. SURFACE AREA 283
35.1. Background 283
35.2. Exercises 284
35.3. Problems 286
35.4. Answers to Odd-Numbered Exercises 287
Chapter 36. TRIPLE INTEGRALS 289
36.1. Background 289
36.2. Exercises 290
36.3. Answers to Odd-Numbered Exercises 293
Chapter 37. CHANGE OF VARIABLES IN AN INTEGRAL 295
37.1. Background 295
37.2. Exercises 296
37.3. Problems 298
37.4. Answers to Odd-Numbered Exercises 299
Chapter 38. VECTOR FIELDS 301
38.1. Background 301
38.2. Exercises 302
38.3. Answers to Odd-Numbered Exercises 304

Part 10. THE CALCULUS OF DIFFERENTIAL FORMS 305
Chapter 39. DIFFERENTIAL FORMS 307
39.1. Background 307
39.2. Exercises 309
39.3. Problems 310
39.4. Answers to Odd-Numbered Exercises 311
Chapter 40. THE EXTERIOR DIFFERENTIAL OPERATOR 313
40.1. Background 313
40.2. Exercises 315
40.3. Problems 316
40.4. Answers to Odd-Numbered Exercises 317
Chapter 41. THE HODGE STAR OPERATOR 319
41.1. Background 319
41.2. Exercises 320

,viii CONTENTS

41.3. Problems 321
41.4. Answers to Odd-Numbered Exercises 322
Chapter 42. CLOSED AND EXACT DIFFERENTIAL FORMS 323
42.1. Background 323
42.2. Exercises 324
42.3. Problems 325
42.4. Answers to Odd-Numbered Exercises 326

Part 11. THE FUNDAMENTAL THEOREM OF CALCULUS 327
Chapter 43. MANIFOLDS AND ORIENTATION 329
43.1. Background—The Language of Manifolds 329
Oriented points 330
Oriented curves 330
Oriented surfaces 330
Oriented solids 331
43.2. Exercises 332
43.3. Problems 334
43.4. Answers to Odd-Numbered Exercises 335
Chapter 44. LINE INTEGRALS 337
44.1. Background 337
44.2. Exercises 338
44.3. Problems 342
44.4. Answers to Odd-Numbered Exercises 343
Chapter 45. SURFACE INTEGRALS 345
45.1. Background 345
45.2. Exercises 346
45.3. Problems 348
45.4. Answers to Odd-Numbered Exercises 349
Chapter 46. STOKES’ THEOREM 351
46.1. Background 351
46.2. Exercises 352
46.3. Problems 356
46.4. Answers to Odd-Numbered Exercises 358
Bibliography 359
Index 361

, Preface

This is a set of exercises and problems for a (more or less) standard beginning calculus sequence.
While a fair number of the exercises involve only routine computations, many of the exercises and
most of the problems are meant to illuminate points that in my experience students have found
confusing.
Virtually all of the exercises have fill-in-the-blank type answers. Often an exercise will end
√ π
with something like, “ . . . so the answer is a 3 + where a = and b = .” One
b
advantage of this type of answer is that it makes it possible to provide students with feedback on a
substantial number of homework exercises without a huge investment of time. More importantly,
it gives students a way of checking their work without giving them the answers. When a student
√ π
works through the exercise and comes up with an answer that doesn’t look anything like a 3 + ,
b
he/she has been given an obvious invitation to check his/her work.
The major drawback of this type of answer is that it does nothing to promote good communi-
cation skills, a matter which in my opinion is of great importance even in beginning courses. That
is what the problems are for. They require logically thought through, clearly organized, and clearly
written up reports. In my own classes I usually assign problems for group work outside of class.
This serves the dual purposes of reducing the burden of grading and getting students involved in
the material through discussion and collaborative work.
This collection is divided into parts and chapters roughly by topic. Many chapters begin with
a “background” section. This is most emphatically not intended to serve as an exposition of the
relevant material. It is designed only to fix notation, definitions, and conventions (which vary
widely from text to text) and to clarify what topics one should have studied before tackling the
exercises and problems that follow.
The flood of elementary calculus texts published in the past half century shows, if nothing else,
that the topics discussed in a beginning calculus course can be covered in virtually any order. The
divisions into chapters in these notes, the order of the chapters, and the order of items within a
chapter is in no way intended to reflect opinions I have about the way in which (or even if) calculus
should be taught. For the convenience of those who might wish to make use of these notes I have
simply chosen what seems to me one fairly common ordering of topics. Neither the exercises nor the
problems are ordered by difficulty. Utterly trivial problems sit alongside ones requiring substantial
thought.
Each chapter ends with a list of the solutions to all the odd-numbered exercises.
The great majority of the “applications” that appear here, as in most calculus texts, are best
regarded as jests whose purpose is to demonstrate in the very simplest ways some connections
between physical quantities (area of a field, volume of a silo, speed of a train, etc.) and the
mathematics one is learning. It does not make these “real world” problems. No one seriously
imagines that some Farmer Jones is really interested in maximizing the area of his necessarily
rectangular stream-side pasture with a fixed amount of fencing, or that your friend Sally just
happens to notice that the train passing her is moving at 54.6 mph. To my mind genuinely
interesting “real world” problems require, in general, way too much background to fit comfortably
into an already overstuffed calculus course. You will find in this collection just a very few serious
applications, problem 15 in Chapter 29, for example, where the background is either minimal or
largely irrelevant to the solution of the problem.
ix

, x PREFACE

I make no claims of originality. While I have dreamed up many of the items included here,
there are many others which are standard calculus exercises that can be traced back, in one form or
another, through generations of calculus texts, making any serious attempt at proper attribution
quite futile. If anyone feels slighted, please contact me.
There will surely be errors. I will be delighted to receive corrections, suggestions, or criticism
at

I have placed the the L TEX source files on my web page so that anyone who wishes can download
A
the material, edit it, add to it, and use it for any noncommercial purpose.