Single and Multivariable Calculus

Contents......Page 5
Introduction......Page 13
1. Analytic Geometry......Page 15
1. Lines......Page 16
2. Distance Between Two Points; Circles......Page 21
3. Functions......Page 22
4. Shifts and Dilations......Page 27
1. The slope of a function......Page 31
2. An example......Page 36
3. Limits......Page 38
4. The Derivative Function......Page 48
5. Adjectives For Functions......Page 53
1. The Power Rule......Page 57
2. Linearity of the Derivative......Page 60
3. The Product Rule......Page 62
4. The Quotient Rule......Page 64
5. The Chain Rule......Page 67
1. Trigonometric Functions......Page 73
2. The Derivative of the sine......Page 76
3. A hard limit......Page 77
4. The Derivative of the sine, continued......Page 80
5. Derivatives of the Trigonometric Functions......Page 81
6. Exponential and Logarithmic functions......Page 82
7. Derivatives of the exponential and logarithmic functions......Page 84
8. Implicit Differentiation......Page 89
9. Inverse Trigonometric Functions......Page 94
10. Limits revisited......Page 97
11. Hyperbolic Functions......Page 102
1. Maxima and Minima......Page 107
2. The first derivative test......Page 111
3. The second derivative test......Page 112
4. Concavity and inflection points......Page 113
5. Asymptotes and Other Things to Look For......Page 115
1. Optimization......Page 119
2. Related Rates......Page 131
3. Newton's Method......Page 139
4. Linear Approximations......Page 143
5. The Mean Value Theorem......Page 145
1. Two examples......Page 149
2. The Fundamental Theorem of Calculus......Page 153
3. Some Properties of Integrals......Page 160
8. Techniques of Integration......Page 165
1. Substitution......Page 166
2. Powers of sine and cosine......Page 171
3. Trigonometric Substitutions......Page 173
4. Integration by Parts......Page 176
5. Rational Functions......Page 180
6. Numerical Integration......Page 184
7. Additional exercises......Page 189
1. Area between curves......Page 191
2. Distance, Velocity, Acceleration......Page 196
3. Volume......Page 199
4. Average value of a function......Page 206
5. Work......Page 209
6. Center of Mass......Page 213
7. Kinetic energy; improper integrals......Page 219
8. Probability......Page 223
9. Arc Length......Page 232
10. Surface Area......Page 234
1. Polar Coordinates......Page 241
2. Slopes in polar coordinates......Page 245
3. Areas in polar coordinates......Page 247
4. Parametric Equations......Page 251
5. Calculus with Parametric Equations......Page 253
11. Sequences and Series......Page 257
1. Sequences......Page 258
2. Series......Page 264
3. The Integral Test......Page 268
4. Alternating Series......Page 273
5. Comparison Tests......Page 275
6. Absolute Convergence......Page 278
7. The Ratio and Root Tests......Page 279
8. Power Series......Page 282
9. Calculus with Power Series......Page 285
10. Taylor Series......Page 287
11. Taylor's Theorem......Page 290
12. Additional exercises......Page 296
1. The Coordinate System......Page 299
2. Vectors......Page 302
3. The Dot Product......Page 307
4. The Cross Product......Page 313
5. Lines and Planes......Page 317
6. Other Coordinate Systems......Page 323
1. Space Curves......Page 329
2. Calculus with vector functions......Page 331
3. Arc length and curvature......Page 339
4. Motion along a curve......Page 345
1. Functions of Several Variables......Page 349
2. Limits and Continuity......Page 353
3. Partial Differentiation......Page 357
4. The Chain Rule......Page 364
5. Directional Derivatives......Page 366
6. Higher order derivatives......Page 371
7. Maxima and minima......Page 373
8. Lagrange Multipliers......Page 378
1. Volume and Average Height......Page 385
2. Double Integrals in Cylindrical Coordinates......Page 395
3. Moment and Center of Mass......Page 399
4. Surface Area......Page 402
5. Triple Integrals......Page 404
6. Cylindrical and Spherical Coordinates......Page 407
7. Change of Variables......Page 411
1. Vector Fields......Page 419
2. Line Integrals......Page 421
3. The Fundamental Theorem of Line Integrals......Page 425
4. Green's Theorem......Page 428
5. Divergence and Curl......Page 433
6. Vector Functions for Surfaces......Page 436
7. Surface Integrals......Page 442
8. Stokes's Theorem......Page 446
9. The Divergence Theorem......Page 450
17. Differential Equations......Page 455
1. First Order Differential Equations......Page 456
2. First Order Homogeneous Linear Equations......Page 460
3. First Order Linear Equations......Page 463
4. Approximation......Page 466
5. Second Order Homogeneous Equations......Page 469
6. Second Order Linear Equations......Page 473
7. Second Order Linear Equations, take two......Page 477
A. Selected Answers......Page 481
B. Useful Formulas......Page 507
Index......Page 511