Calculus 11th Edition.pdf

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Contents
P Preparation for Calculus 1
P.1 Graphs and Models 2
P.2 Linear Models and Rates of Change 10
P.3 Functions and Their Graphs 19
P.4 Review of Trigonometric Functions 31
Review Exercises 41
P.S. Problem Solving 43
1 Limits and Their Properties 45
1.1 A Preview of Calculus 46
1.2 Finding Limits Graphically and Numerically 52
1.3 Evaluating Limits Analytically 63
1.4 Continuity and One-Sided Limits 74
1.5 Infinite Limits 87
 Section Project: Graphs and Limits of
Trigonometric Functions 94
Review Exercises 95
P.S. Problem Solving 97
2 Differentiation 99
2.1 The Derivative and the Tangent Line Problem 100
2.2 Basic Differentiation Rules and Rates of Change 110
2.3 Product and Quotient Rules and Higher-Order
 Derivatives 122
2.4 The Chain Rule 133
2.5 Implicit Differentiation 144
Section Project: Optical Illusions 151
2.6 Related Rates 152
Review Exercises 161
P.S. Problem Solving 163
3 Applications of Differentiation 165
3.1 Extrema on an Interval 166
3.2 Rolle’s Theorem and the Mean Value Theorem 174
3.3 Increasing and Decreasing Functions and
the First Derivative Test 181
Section Project: Even Fourth-Degree Polynomials 190
3.4 Concavity and the Second Derivative Test 191
3.5 Limits at Infinity 199
3.6 A Summary of Curve Sketching 209
3.7 Optimization Problems 219
Section Project: Minimum Time 228
3.8 Newton’s Method 229
3.9 Differentials 235
Review Exercises 242
P.S. Problem Solving 245
iv Contents
4 Integration 247
4.1 Antiderivatives and Indefinite Integration 248
4.2 Area 258
4.3 Riemann Sums and Definite Integrals 270
4.4 The Fundamental Theorem of Calculus 281
 Section Project: Demonstrating the
Fundamental Theorem 295
4.5 Integration by Substitution 296
Review Exercises 309
P.S. Problem Solving 311
5 Logarithmic, Exponential, and
Other Transcendental Functions 313
5.1 The Natural Logarithmic Function: Differentiation 314
5.2 The Natural Logarithmic Function: Integration 324
5.3 Inverse Functions 333
5.4 Exponential Functions: Differentiation and Integration 342
5.5 Bases Other than e and Applications 352
 Section Project: Using Graphing Utilities to
Estimate Slope 361
5.6 Indeterminate Forms and L’Hôpital’s Rule 362
5.7 Inverse Trigonometric Functions: Differentiation 373
5.8 Inverse Trigonometric Functions: Integration 382
5.9 Hyperbolic Functions 390
Section Project: Mercator Map 399
Review Exercises 400
P.S. Problem Solving 403
6 Differential Equations 405
6.1 Slope Fields and Euler’s Method 406
6.2 Growth and Decay 415
6.3 Separation of Variables and the Logistic Equation 423
6.4 First-Order Linear Differential Equations 432
Section Project: Weight Loss 438
Review Exercises 439
P.S. Problem Solving 441
7 Applications of Integration 443
7.1 Area of a Region Between Two Curves 444
7.2 Volume: The Disk Method 454
7.3 Volume: The Shell Method 465
Section Project: Saturn 473
7.4 Arc Length and Surfaces of Revolution 474
7.5 Work 485
Section Project: Pyramid of Khufu 493
7.6 Moments, Centers of Mass, and Centroids 494
7.7 Fluid Pressure and Fluid Force 505
Review Exercises 511
P.S. Problem Solving 513
Contents v
8 Integration Techniques and Improper Integrals 515
8.1 Basic Integration Rules 516
8.2 Integration by Parts 523
8.3 Trigonometric Integrals 532
Section Project: The Wallis Product 540
8.4 Trigonometric Substitution 541
8.5 Partial Fractions 550
8.6 Numerical Integration 559
8.7 Integration by Tables and Other Integration Techniques 566
8.8 Improper Integrals 572
Review Exercises 583
P.S. Problem Solving 585
9 Infinite Series 587
9.1 Sequences 588
9.2 Series and Convergence 599
Section Project: Cantor’s Disappearing Table 608
9.3 The Integral Test and p-Series 609
Section Project: The Harmonic Series 615
9.4 Comparisons of Series 616
9.5 Alternating Series 623
9.6 The Ratio and Root Tests 631
9.7 Taylor Polynomials and Approximations 640
9.8 Power Series 651
9.9 Representation of Functions by Power Series 661
9.10 Taylor and Maclaurin Series 668
Review Exercises 680
P.S. Problem Solving 683
10 Conics, Parametric Equations, and
Polar Coordinates 685
10.1 Conics and Calculus 686
10.2 Plane Curves and Parametric Equations 700
Section Project: Cycloids 709
10.3 Parametric Equations and Calculus 710
10.4 Polar Coordinates and Polar Graphs 719
Section Project: Cassini Oval 728
10.5 Area and Arc Length in Polar Coordinates 729
10.6 Polar Equations of Conics and Kepler’s Laws 738
Review Exercises 746
P.S. Problem Solving 749
vi Contents
11 Vectors and the Geometry of Space 751
11.1 Vectors in the Plane 752
11.2 Space Coordinates and Vectors in Space 762
11.3 The Dot Product of Two Vectors 770
11.4 The Cross Product of Two Vectors in Space 779
11.5 Lines and Planes in Space 787
Section Project: Distances in Space 797
11.6 Surfaces in Space 798
11.7 Cylindrical and Spherical Coordinates 808
Review Exercises 815
P.S. Problem Solving 817
12 Vector-Valued Functions 819
12.1 Vector-Valued Functions 820
Section Project: Witch of Agnesi 827
12.2 Differentiation and Integration of Vector-Valued
 Functions 828
12.3 Velocity and Acceleration 836
12.4 Tangent Vectors and Normal Vectors 845
12.5 Arc Length and Curvature 855
Review Exercises 867
P.S. Problem Solving 869
13 Functions of Several Variables 871
13.1 Introduction to Functions of Several Variables 872
13.2 Limits and Continuity 884
13.3 Partial Derivatives 894
13.4 Differentials 904
13.5 Chain Rules for Functions of Several Variables 911
13.6 Directional Derivatives and Gradients 919
13.7 Tangent Planes and Normal Lines 931
Section Project: Wildflowers 939
13.8 Extrema of Functions of Two Variables 940
13.9 Applications of Extrema 948
Section Project: Building a Pipeline 955
13.10 Lagrange Multipliers 956
Review Exercises 964
P.S. Problem Solving 967
14 Multiple Integration 969
14.1 Iterated Integrals and Area in the Plane 970
14.2 Double Integrals and Volume 978
14.3 Change of Variables: Polar Coordinates 990
14.4 Center of Mass and Moments of Inertia 998
Section Project: Center of Pressure on a Sail 1005
14.5 Surface Area 1006
Section Project: Surface Area in Polar Coordinates 1012
14.6 Triple Integrals and Applications 1013
14.7 Triple Integrals in Other Coordinates 1024
Section Project: Wrinkled and Bumpy Spheres 1030
14.8 Change of Variables: Jacobians 1031
Review Exercises 1038
P.S. Problem Solving 1041
Contents vii
15 Vector Analysis 1043
15.1 Vector Fields 1044
15.2 Line Integrals 1055
15.3 Conservative Vector Fields and Independence of Path 1069
15.4 Green’s Theorem 1079
Section Project: Hyperbolic and Trigonometric Functions 1087
15.5 Parametric Surfaces 1088
15.6 Surface Integrals 1098
Section Project: Hyperboloid of One Sheet 1109
15.7 Divergence Theorem 1110
15.8 Stokes’s Theorem 1118
Review Exercises 1124
P.S. Problem Solving 1127
16 Additional Topics in Differential Equations (Online)*
16.1 Exact First-Order Equations
16.2 Second-Order Homogeneous Linear Equations
16.3 Second-Order Nonhomogeneous Linear Equations
Section Project: Parachute Jump
16.4 Series Solutions of Differential Equations
Review Exercises
P.S. Problem Solving
Appendices
Appendix A: Proofs of Selected Theorems A2
Appendix B: Integration Tables A3
Appendix C: Precalculus Review (Online)*
Appendix D: Rotation and the General Second-Degree
Equation (Online)*
Appendix E: Complex Numbers (Online)*
Appendix F: Business and Economic Applications (Online)*
Appendix G: Fitting Models to Data (Online)*
Answers to All Odd-Numbered Exercises A7
 Index A121