Idiot's Guides - Calculus II

Idiots Guides: Calculus II, like its counterpart Idiots Guides: Calculus I, is a curriculum-based companion book that continues the tradition of taking the sting out of calculus by adding more explanatory graphs and illustrations in easy-to-understand language, practice problems, and even a test at the end. Idiots Guides: Calculus II is geared for all students who need to succeed in calculus. Also included: • Complete step-by-step examples to help you work through the problems. • Advanced and complex problem examples. • Sidebar problems sprinkled throughout to test readers knowledge with answer key in the back. • Practice test included at the end of the book, complete with answer key. Contents iii 5 Part 1: Review of Pre-Calculus and Calculus I 1 13 1 Pre-Calculus Topics Used in Calculus II 3 15 Trigonometry 4 16 Exponents and Logarithms 6 18 Parametric Equations 8 20 Polar Coordinates 10 22 Geometric Sequences and Series 14 26 Partial Fractions 15 27 2 Limits, Derivatives, and Basic Integration 19 31 Limits 20 32 Derivatives 23 35 Implicit Differentiation 28 40 Mean Value Theorem 29 41 Relative Extremes and Concavity 30 42 Applications of the Derivative 33 45 Related Rates 36 48 3 Definite and Indefinite Integrals 41 53 Indefinite Integrals 42 54 u-Substitutions 46 58 Fundamental Theorem of Calculus 50 62 Numerical Approximation 54 66 Part 2: Length, Area, and Volumes 61 73 4 Areas and Approximations 63 75 Riemann Sums 64 76 From Numerical Approximations to the True Area 76 88 Average Value of a Function 80 92 Area Between Two Curves 81 93 Simpson’s Rule 86 98 5 Volumes and Areas of Solids of Revolutions 91 103 Volumes of Solids with Defined Cross Sections 92 104 Disks and Washers 97 109 Cylindrical Shell Method 106 118 Arc Length 110 122 Surface Area 113 125 Part 3: More Definite and Indefinite Integrals 117 129 6 More Integration Techniques 119 131 Integration by Parts 120 132 Polynomials and Transcendentals 123 135 Two Transcendentals 129 141 7 Integration with Trigonometric Functions 133 145 Trigonometric Substitutions 134 146 Case I: a2 + x2 134 146 Case II: x2 - a2 136 148 Case III: a2 - x2 138 150 Integrals of the Form sinn(x) cosm(x) (When Either m or n Is Odd) 139 151 Case I: Both m and n Are Odd 139 151 Case II: Both m and n Are Even 140 152 Case III: Either m or n Is Odd 141 153 Integrals with Integrands of the Form tann(x) secm(x) (m Is Even) 142 154 Integrals with Integrands of the Form tann(x) secm(x) (m Is Odd, n Is Even) 143 155 8 Integration with Fractions 147 159 Completing the Square 148 160 Integration by Partial Fractions 152 164 Nonrepeating Linear Factors 152 164 Repeated Linear Factors 154 166 Irreducible Quadratic Factors 155 167 Part 4: The Infinite Series and More 159 171 9 To Infinity and Beyond 161 173 Improper Integrals 162 174 Infinite Limits of Integration 162 174 Discontinuities in the Integrand 165 177 Comparison Test for Improper Integrals 168 180 10 Parametric Equations 171 183 First and Second Derivatives of Parametric Curves 172 184 Arc Length of a Parametric Curve 178 190 11 Polar Coordinates 181 193 Slope of the Tangent Line 182 194 Length of an Arc of a Polar Curve 187 199 Area Under a Curve 189 201 12 Introduction to Vectors 195 207 Scalars and Vectors 196 208 13 Differential Equations 205 217 Separable Differential Equations 206 218 Linear Approximations 212 224 Euler’s Method 214 226 Slope Fields 217 229 First Order Linear Differential Equations 222 234 14 Infinite Sequences 227 239 Convergence and Divergence of Sequences 228 240 Squeeze Theorem 231 243 Increasing, Decreasing, and Monotonic Sequences 233 245 15 Infinite Series 237 249 Infinite Geometric Series 238 250 Tests of Convergence 239 251 Alternating Series 244 256 Estimating the Sum of Alternating Series 247 259 16 Power Series 249 261 Power Series 250 262 MacLaurin Series 252 264 Taylor Series 258 270 Error Estimates for the MacLaurin and Taylor Series 260 272 17 Calculus II Final Exam 263 275 Chapter 1 264 276 Chapter 2 264 276 Chapter 3 265 277 Chapter 4 266 278 Chapter 5 266 278 Chapter 6 267 279 Chapter 7 267 279 Chapter 8 268 280 Chapter 9 268 280 Chapter 10 268 280 Chapter 11 269 281 Chapter 12 269 281 Chapter 13 270 282 Chapter 14 270 282 Chapter 15 271 283 Chapter 16 271 283 Solutions 272 284 Chapter 1 272 284 Chapter 2 272 284 Chapter 3 273 285 Chapter 4 273 285 Chapter 5 274 286 Chapter 6 274 286 Chapter 7 274 286 Chapter 8 275 287 Chapter 9 275 287 Chapter 10 275 287 Chapter 11 275 287 Chapter 12 276 288 Chapter 13 276 288 Chapter 14 277 289 Chapter 15 277 289 Chapter 16 277 289 A Solutions to “You’ve Got Problems” 279 291 B Integration Practice Problems and Solutions 297 309 C Glossary 307 319 Index 313 325