Exercises and Problems in Calculus John M. Erdman Portland State University Version August 1, 2013 c 2010 John M. Erdman E-mail address: [email protected] Contents Preface ix Part 1. PRELIMINARY MATERIAL 1 Chapter 1. INEQUALITIES AND ABSOLUTE VALUES3 1.1. Background 3 1.2. Exercises 4 1.3. Problems 5 1.4. Answers to Odd-Numbered Exercises6 Chapter 2. LINES IN THE PLANE7 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'HOPITAL'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 n Chapter 22. VECTOR AND METRIC PROPERTIES of R 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 p 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 p 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.