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Contents iii
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
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
12 Introduction to Vectors 195
Scalars and Vectors 196
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
A Solutions to “You’ve Got Problems” 279
B Integration Practice Problems and Solutions 297
C Glossary 307
Index 313
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