Calculus

Cover......Page 1
Contents......Page 11
Preface......Page 16
Additional Resources......Page 19
Acknowledgements......Page 21
P: Preparation for Calculus......Page 25
P.1 Graphs and Models......Page 26
P.1 Exercises......Page 32
P.2 Linear Models and Rates of Change......Page 34
P.2 Exercises......Page 40
P.3 Functions and Their Graphs......Page 43
P.3 Exercises......Page 51
P.4 Fitting Models to Data......Page 55
P.4 Exercises......Page 58
Review Exercises......Page 61
P.S. Problem Solving......Page 63
Ch 1: Limits and Their Properties......Page 65
1.1 A Preview of Calculus......Page 66
1.1 Exercises......Page 71
1.2 Finding Limits Graphically and Numerically......Page 72
1.2 Exercises......Page 79
1.3 Evaluating Limits Analytically......Page 83
1.3 Exercises......Page 91
1.4 Continuity and One-Sided Limits......Page 94
1.4 Exercises......Page 103
1.5 Infinite Limits......Page 107
1.5 Exercises......Page 112
Review Exercises......Page 115
P.S. Problem Solving......Page 117
Ch 2: Differentiation......Page 119
2.1 The Derivative and the Tangent Line Problem......Page 120
2.1 Exercises......Page 127
2.2 Basic Differentiation Rules and Rates of Change......Page 130
2.2 Exercises......Page 138
2.3 Product and Quotient Rules and Higher-Order Derivatives......Page 142
2.3 Exercises......Page 149
2.4 The Chain Rule......Page 153
2.4 Exercises......Page 160
2.5 Implicit Differentiation......Page 164
2.5 Exercises......Page 169
2.6 Related Rates......Page 172
2.6 Exercises......Page 177
Review Exercises......Page 181
P.S. Problem Solving......Page 183
Ch 3: Applications of Differentiation......Page 185
3.1 Extrema on an Interval......Page 186
3.1 Exercises......Page 191
3.2 Rolle’s Theorem and the Mean Value Theorem......Page 194
3.2 Exercises......Page 198
3.3 Increasing and Decreasing Functions and the First Derivative Test......Page 201
3.3 Exercises......Page 207
3.4 Concavity and the Second Derivative Test......Page 211
3.4 Exercises......Page 216
3.5 Limits at Infinity......Page 219
3.5 Exercises......Page 226
3.6 A Summary of Curve Sketching......Page 230
3.6 Exercises......Page 236
3.7 Optimization Problems......Page 239
3.7 Exercises......Page 244
3.8 Newton’s Method......Page 249
3.8 Exercises......Page 253
3.9 Differentials......Page 255
3.9 Exercises......Page 260
Review Exercises......Page 262
P.S. Problem Solving......Page 265
Ch 4: Integration......Page 267
4.1 Antiderivatives and Indefinite Integration......Page 268
4.1 Exercises......Page 275
4.2 Area......Page 278
4.2 Exercises......Page 287
4.3 Riemann Sums and Definite Integrals......Page 290
4.3 Exercises......Page 297
4.4 The Fundamental Theorem of Calculus......Page 301
4.4 Exercises......Page 312
4.5 Integration by Substitution......Page 316
4.5 Exercises......Page 325
4.6 Numerical Integration......Page 329
4.6 Exercises......Page 334
Review Exercises......Page 336
P.S. Problem Solving......Page 339
Ch 5: Logarithmic, Exponential, and Other Transcendental Functions......Page 341
5.1 The Natural Logarithmic Function: Differentiation......Page 342
5.1 Exercises......Page 349
5.2 The Natural Logarithmic Function: Integration......Page 352
5.2 Exercises......Page 358
5.3 Inverse Functions......Page 361
5.3 Exercises......Page 367
5.4 Exponential Functions: Differentiation and Integration......Page 370
5.4 Exercises......Page 376
5.5 Bases Other than e and Applications......Page 380
5.5 Exercises......Page 386
5.6 Inverse Trigonometric Functions: Differentiation......Page 390
5.6 Exercises......Page 396
5.7 Inverse Trigonometric Functions: Integration......Page 399
5.7 Exercises......Page 404
5.8 Hyperbolic Functions......Page 407
5.8 Exercises......Page 414
Review Exercises......Page 417
P.S. Problem Solving......Page 419
Ch 6: Differential Equations......Page 421
6.1 Slope Fields and Euler’s Method......Page 422
6.1 Exercises......Page 427
6.2 Differential Equations: Growth and Decay......Page 431
6.2 Exercises......Page 436
6.3 Separation of Variables and the Logistic Equation......Page 439
6.3 Exercises......Page 445
6.4 First-Order Linear Differential Equations......Page 448
6.4 Exercises......Page 452
Review Exercises......Page 455
P.S. Problem Solving......Page 457
Ch 7: Applications of Integration......Page 459
7.1 Area of a Region Between Two Curves......Page 460
7.1 Exercises......Page 466
7.2 Volume:The Disk Method......Page 470
7.2 Exercises......Page 477
7.3 Volume: The Shell Method......Page 481
7.3 Exercises......Page 486
7.4 Arc Length and Surfaces of Revolution......Page 490
7.4 Exercises......Page 497
7.5 Work......Page 501
7.5 Exercises......Page 507
7.6 Moments, Centers of Mass, and Centroids......Page 510
7.6 Exercises......Page 518
7.7 Fluid Pressure and Fluid Force......Page 521
7.7 Exercises......Page 525
Review Exercises......Page 527
P.S. Problem Solving......Page 529
Ch 8: Integration Techniques, L’Hôpital’s Rule, and Improper Integrals......Page 531
8.1 Basic Integration Rules......Page 532
8.1 Exercises......Page 536
8.2 Integration by Parts......Page 539
8.2 Exercises......Page 545
8.3 Trigonometric Integrals......Page 548
8.3 Exercises......Page 554
8.4 Trigonometric Substitution......Page 557
8.4 Exercises......Page 563
8.5 Partial Fractions......Page 566
8.5 Exercises......Page 573
8.6 Integration by Tables and Other Integration Techniques......Page 575
8.6 Exercises......Page 579
8.7 Indeterminate Forms and L’Hôpital’s Rule......Page 581
8.7 Exercises......Page 588
8.8 Improper Integrals......Page 592
8.8 Exercises......Page 599
Review Exercises......Page 603
P.S. Problem Solving......Page 605
Ch 9: Infinite Series......Page 607
9.1 Sequences......Page 608
9.1 Exercises......Page 616
9.2 Series and Convergence......Page 619
9.2 Exercises......Page 625
9.3 The Integral Test and p-Series......Page 629
9.3 Exercises......Page 633
9.4 Comparisons of Series......Page 636
9.4 Exercises......Page 640
9.5 Alternating Series......Page 643
9.5 Exercises......Page 649
9.6 The Ratio and Root Tests......Page 651
9.6 Exercises......Page 657
9.7 Taylor Polynomials and Approximations......Page 660
9.7 Exercises......Page 668
9.8 Power Series......Page 671
9.8 Exercises......Page 678
9.9 Representation of Functions by Power Series......Page 681
9.9 Exercises......Page 686
9.10 Taylor and Maclaurin Series......Page 688
9.10 Exercises......Page 697
Review Exercises......Page 700
P.S. Problem Solving......Page 703
Ch 10: Conics, Parametric Equations, and Polar Coordinates......Page 705
10.1 Conics and Calculus......Page 706
10.1 Exercises......Page 716
10.2 Plane Curves and Parametric Equations......Page 720
10.2 Exercises......Page 727
10.3 Parametric Equations and Calculus......Page 730
10.3 Exercises......Page 735
10.4 Polar Coordinates and Polar Graphs......Page 739
10.4 Exercises......Page 746
10.5 Area and Arc Length in Polar Coordinates......Page 749
10.5 Exercises......Page 755
10.6 Polar Equations of Conics and Kepler’s Laws......Page 758
10.6 Exercises......Page 763
Review Exercises......Page 766
P.S. Problem Solving......Page 769
Ch 11: Vectors and the Geometry of Space......Page 771
11.1 Vectors in the Plane......Page 772
11.1 Exercises......Page 779
11.2 Space Coordinates and Vectors in Space......Page 782
11.2 Exercises......Page 787
11.3 The Dot Product of Two Vectors......Page 790
11.3 Exercises......Page 797
11.4 The Cross Product of Two Vectors in Space......Page 799
11.4 Exercises......Page 805
11.5 Lines and Planes in Space......Page 807
11.5 Exercises......Page 814
11.6 Surfaces in Space......Page 818
11.6 Exercises......Page 826
11.7 Cylindrical and Spherical Coordinates......Page 828
11.7 Exercises......Page 833
Review Exercises......Page 835
P.S. Problem Solving......Page 837
Ch 12: Vector-Valued Functions......Page 839
12.1 Vector-Valued Functions......Page 840
12.1 Exercises......Page 845
12.2 Differentiation and Integration of Vector-Valued Functions......Page 848
12.2 Exercises......Page 854
12.3 Velocity and Acceleration......Page 856
12.3 Exercises......Page 862
12.4 Tangent Vectors and Normal Vectors......Page 865
12.4 Exercises......Page 872
12.5 Arc Length and Curvature......Page 875
12.5 Exercises......Page 884
Review Exercises......Page 887
P.S. Problem Solving......Page 889
Ch 13: Functions of Several Variables......Page 891
13.1 Introduction to Functions of Several Variables......Page 892
13.1 Exercises......Page 900
13.2 Limits and Continuity......Page 904
13.2 Exercises......Page 911
13.3 Partial Derivatives......Page 914
13.3 Exercises......Page 920
13.4 Differentials......Page 924
13.4 Exercises......Page 929
13.5 Chain Rules for Functions of Several Variables......Page 931
13.5 Exercises......Page 937
13.6 Directional Derivatives and Gradients......Page 939
13.6 Exercises......Page 948
13.7 Tangent Planes and Normal Lines......Page 951
13.7 Exercises......Page 957
13.8 Extrema of Functions of Two Variables......Page 960
13.8 Exercises......Page 966
13.9 Applications of Extrema......Page 968
13.9 Exercises......Page 973
13.10 Lagrange Multipliers......Page 976
13.10 Exercises......Page 982
Review Exercises......Page 984
P.S. Problem Solving......Page 987
Ch 14: Multiple Integration......Page 989
14.1 Iterated Integrals and Area in the Plane......Page 990
14.1 Exercises......Page 996
14.2 Double Integrals and Volume......Page 998
14.2 Exercises......Page 1007
14.3 Change of Variables: Polar Coordinates......Page 1010
14.3 Exercises......Page 1015
14.4 Center of Mass and Moments of Inertia......Page 1018
14.4 Exercises......Page 1024
14.5 Surface Area......Page 1026
14.5 Exercises......Page 1031
14.6 Triple Integrals and Applications......Page 1033
14.6 Exercises......Page 1041
14.7 Triple Integrals in Other Coordinates......Page 1044
14.7 Exercises......Page 1049
14.8 Change of Variables: Jacobians......Page 1051
14.8 Exercises......Page 1056
Review Exercises......Page 1058
P.S. Problem Solving......Page 1061
Ch 15: Vector Analysis......Page 1063
15.1 Vector Fields......Page 1064
15.1 Exercises......Page 1073
15.2 Line Integrals......Page 1075
15.2 Exercises......Page 1085
15.3 Conservative Vector Fields and Independence of Path......Page 1089
15.3 Exercises......Page 1096
15.4 Green’s Theorem......Page 1099
15.4 Exercises......Page 1105
15.5 Parametric Surfaces......Page 1108
15.5 Exercises......Page 1115
15.6 Surface Integrals......Page 1118
15.6 Exercises......Page 1128
15.7 Divergence Theorem......Page 1130
15.7 Exercises......Page 1136
15.8 Stokes’s Theorem......Page 1138
15.8 Exercises......Page 1143
Review Exercises......Page 1144
P.S. Problem Solving......Page 1147
Appendices......Page 1149
A: Proofs of Selected Theorems......Page 1150
B: Integration Tables......Page 1151
Answers to Odd-Numbered Exercises......Page 1155
Index......Page 1263