Calculus: One and Several Variables, 10th Edition

Cover Page\r......Page 1
Title Page\r......Page 7
Copyright Page\r......Page 8
Preface\r......Page 9
Contents......Page 17
1.1 What is Calculus?\r......Page 25
1.2 Review of Elementary Mathematics\r......Page 27
1.3 Review of Inequalities\r......Page 35
1.4 Coordinate Plane; Analytic Geometry\r......Page 41
1.5 Functions\r......Page 48
1.6 The Elementary Functions\r......Page 56
1.7 Combinations of Functions\r......Page 65
1.8 A Note on Mathematical Proof; Mathematical Induction\r......Page 71
2.1 The Limit Process (An Intuitive Introduction)\r......Page 77
2.2 Definition of Limit\r......Page 88
2.3 Some Limit Theorems\r......Page 97
2.4 Continuity\r......Page 106
2.5 The Pinching Theorem; Trigonometric Limits\r......Page 115
2.6 Two Basic Theorems\r......Page 121
Project 2.6 The Bisection Method for Finding the Roots of f (x) = 0\r......Page 126
3.1 The Derivative\r......Page 129
3.2 Some Differentiation Formulas\r......Page 139
3.3 The d/dx Notation; Derivatives of Higher Order\r......Page 148
3.4 The Derivative As A Rate of Change\r......Page 154
3.5 The Chain Rule\r......Page 157
3.6 Differentiating The Trigonometric Functions\r......Page 166
3.7 Implicit Differentiation; Rational Powers\r......Page 171
4.1 The Mean-Value Theorem\r......Page 178
4.2 Increasing and Decreasing Functions\r......Page 184
4.3 Local Extreme Values\r......Page 191
4.4 Endpoint Extreme Values; Absolute Extreme Values\r......Page 198
4.5 Some Max-Min Problems\r......Page 206
4.6 Concavity and Points of Inflection\r......Page 214
4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps\r......Page 219
4.8 Some Curve Sketching\r......Page 225
4.9 Velocity and Acceleration; Speed\r......Page 233
Project 4.9B Energy of a Falling Body (Near the Surface of the Earth)\r......Page 241
4.10 Related Rates of Change per Unit Time\r......Page 242
4.11 Differentials\r......Page 247
Project 4.11 Marginal Cost, Marginal Revenue, Marginal Profit\r......Page 252
4.12 Newton-Raphson Approximations\r......Page 253
5.1 An Area Problem; a Speed-Distance Problem\r......Page 258
5.2 The Definite Integral of a Continuous Function\r......Page 261
5.3 The Function f (x) =∫xa f(t) dt\r......Page 270
5.4 The Fundamental Theorem of Integral Calculus\r......Page 278
5.5 Some Area Problems\r......Page 284
Project 5.5 Integrability; Integrating Discontinuous Functions\r......Page 290
5.6 Indefinite Integrals\r......Page 292
5.7 Working Back from the Chain Rule; the u-Substitution\r......Page 298
5.8 Additional Properties of the Definite Integral\r......Page 305
5.9 Mean-Value Theorems for Integrals; Average Value of a Function\r......Page 309
6.1 More on Area\r......Page 316
6.2 Volume by Parallel Cross Sections; Disks and Washers\r......Page 320
6.3 Volume by the Shell Method\r......Page 330
6.4 The Centroid of a Region; Pappus’s Theorem on Volumes\r......Page 336
6.5 The Notion of Work\r......Page 343
*6.6 Fluid Force\r......Page 351
7.1 One-to-One Functions; Inverses\r......Page 357
7.2 The Logarithm Function, Part I\r......Page 366
7.3 The Logarithm Function, Part II\r......Page 371
7.4 The Exponential Function\r......Page 380
7.5 Arbitrary Powers; Other Bases\r......Page 388
7.6 Exponential Growth and Decay\r......Page 394
7.7 The Inverse Trigonometric Functions\r......Page 402
Project 7.7 Refraction\r......Page 411
7.8 The Hyperbolic Sine and Cosine\r......Page 412
*7.9 The Other Hyperbolic Functions\r......Page 416
8.1 Integral Tables and Review\r......Page 422
8.2 Integration by Parts\r......Page 426
Project 8.2 Sine Waves y = sin nx and Cosine Waves y = cos nx\r......Page 434
8.3 Powers and Products of Trigonometric Functions\r......Page 435
8.4 Integrals Featuring √(a2-x2),√(a2+x2),√(x2-a2)\r......Page 441
8.5 Rational Functions; Partial Fractions\r......Page 446
*8.6 Some Rationalizing Substitutions\r......Page 454
8.7 Numerical Integration\r......Page 457
CHAPTER 9 SOME DIFFERENTIAL EQUATIONS\r......Page 467
9.1 First-Order Linear Equations\r......Page 468
9.2 Integral Curves; Separable Equations\r......Page 475
Project 9.2 Orthogonal Trajectories\r......Page 482
9.3 The Equation y\"+ay\'+by = 0\r......Page 483
10.1 Geometry of Parabola, Ellipse, Hyperbola\r......Page 493
10.2 Polar Coordinates\r......Page 502
10.3 Sketching Curves in Polar Coordinates\r......Page 508
Project 10.3 Parabola, Ellipse, Hyperbola in Polar Coordinates\r......Page 515
10.4 Area in Polar Coordinates\r......Page 516
10.5 Curves Given Parametrically\r......Page 520
10.6 Tangents to Curves Given Parametrically\r......Page 527
10.7 Arc Length and Speed\r......Page 533
10.8 The Area of A Surface of Revolution; The Centroid of a Curve; Pappus’s Theorem on Surface Area\r......Page 541
Project 10.8 The Cycloid\r......Page 549
11.1 The Least Upper Bound Axiom\r......Page 552
11.2 Sequences of Real Numbers\r......Page 556
11.3 Limit of a Sequence\r......Page 562
Project 11.3 Sequences and the Newton-Raphson Method\r......Page 571
11.4 Some Important Limits\r......Page 574
11.5 The Indeterminate Form (0/0)\r......Page 578
11.6 The Indeterminate Form (∞/∞); Other Indeterminate Forms\r......Page 584
11.7 Improper Integrals\r......Page 589
12.1 Sigma Notation\r......Page 599
12.2 Infinite Series\r......Page 601
12.3 The Integral Test; Basic Comparison, Limit Comparison\r......Page 609
12.4 The Root Test; the Ratio Test\r......Page 617
12.5 Absolute Convergence and Conditional Convergence; Alternating Series\r......Page 621
12.6 Taylor Polynomials in x; Taylor Series in x\r......Page 626
12.7 Taylor Polynomials and Taylor Series in x − a\r......Page 637
12.8 Power Series\r......Page 640
12.9 Differentiation and Integration of Power Series\r......Page 647
Project 12.9A The Binomial Series\r......Page 657
Project 12.9B Estimating π\r......Page 658
13.1 Rectangular Space Coordinates\r......Page 662
13.2 Vectors in Three-Dimensional Space\r......Page 668
13.3 The Dot Product\r......Page 677
13.4 The Cross Product\r......Page 687
13.5 Lines\r......Page 695
13.6 Planes\r......Page 703
Project 13.6 Some Geometry by Vector Methods\r......Page 712
13.7 Higher Dimensions\r......Page 713
CHAPTER 14 VECTOR CALCULUS\r......Page 716
14.1 Limit, Continuity, Vector Derivative\r......Page 718
14.2 The Rules of Differentiation\r......Page 725
14.3 Curves\r......Page 729
14.4 Arc Length\r......Page 738
Project 14.4 More General Changes of Parameter\r......Page 745
14.5 Curvilinear Motion; Curvature\r......Page 747
Project 14.5A Transition Curves\r......Page 756
14.6 Vector Calculus in Mechanics\r......Page 757
*14.7 Planetary Motion\r......Page 765
15.1 Elementary Examples\r......Page 772
15.2 A Brief Catalogue of the Quadric Surfaces; Projections\r......Page 775
15.3 Graphs; Level Curves and Level Surfaces\r......Page 782
Project 15.3 Level Curves and Surfaces\r......Page 790
15.4 Partial Derivatives\r......Page 791
15.5 Open and Closed Sets\r......Page 798
15.6 Limits and Continuity; Equality of Mixed Partials\r......Page 801
Project 15.6 Partial Differential Equations\r......Page 809
16.1 Differentiability and Gradient\r......Page 812
16.2 Gradients and Directional Derivatives\r......Page 820
16.3 The Mean-Value Theorem; the Chain Rule\r......Page 829
16.4 The Gradient as a Normal; Tangent Lines and Tangent Planes\r......Page 842
16.5 Local Extreme Values\r......Page 852
16.6 Absolute Extreme Values\r......Page 860
16.7 Maxima and Minima with Side Conditions\r......Page 865
16.8 Differentials\r......Page 873
16.9 Reconstructing a Function from Its Gradient\r......Page 879
17.1 Multiple-Sigma Notation\r......Page 888
17.2 Double Integrals\r......Page 891
17.3 The Evaluation of Double Integrals by Repeated Integrals\r......Page 902
17.4 The Double Integral as the Limit of Riemann Sums; Polar Coordinates\r......Page 912
17.5 Further Applications of the Double Integral\r......Page 919
17.6 Triple Integrals\r......Page 926
17.7 Reduction to Repeated Integrals\r......Page 931
17.8 Cylindrical Coordinates\r......Page 940
17.9 The Triple Integral as the Limit of Riemann Sums; Spherical Coordinates\r......Page 946
17.10 Jacobians; Changing Variables in Multiple Integration\r......Page 954
Project 17.10 Generalized Polar Coordinates\r......Page 959
18.1 Line Integrals\r......Page 962
18.2 The Fundamental Theorem for Line Integrals\r......Page 970
18.3 Work-Energy Formula; Conservation of Mechanical Energy\r......Page 975
18.4 Another Notation for Line Integrals; Line Integrals with Respect to Arc Length\r......Page 978
18.5 Green’s Theorem\r......Page 983
18.6 Parametrized Surfaces; Surface Area\r......Page 993
18.7 Surface Integrals\r......Page 1004
18.8 The Vector Differential Operator\r......Page 1013
18.9 The Divergence Theorem\r......Page 1019
Project 18.9 Static Charges\r......Page 1024
18.10 Stokes’s Theorem\r......Page 1025
19.1 Bernoulli Equations; Homogeneous Equations\r......Page 1034
19.2 Exact Differential Equations; Integrating Factors\r......Page 1037
19.3 Numerical Methods\r......Page 1042
19.4 The Equation y\"+ay+by = φ(x)\r......Page 1046
19.5 Mechanical Vibrations\r......Page 1054
A.1 Rotation of Axes; Eliminating the xy-Term......Page 1065
A.2 Determinants\r......Page 1067
B.1 The Intermediate-Value Theorem\r......Page 1072
B.2 Boundedness; Extreme-Value Theorem\r......Page 1073
B.3 Inverses\r......Page 1074
B.4 The Integrability of Continuous Functions\r......Page 1075
B.5 The Integral as the Limit of Riemann Sums\r......Page 1078
ANSWERS TO ODD-NUMBERED EXERCISES\r......Page 1079
Index\r......Page 1155
Table of Integrals Inside Covers\r......Page 1167