Calculus with analytic geometry

✍ Scribed by Robert Ellis, Denny Gulick.

• Publisher: Harcourt Brace Jovanovich,
• Year: 1978
• Tongue: English
• Leaves: 1091
• Category: Library

✦ Synopsis

SSID 40291695, A grayscale scanned copy with some figures not portrayed clearly.
Mentioned in this video: https://www.youtube.com/watch?v=u1rRF0lP0Ws&ab_channel=TheMathSorcerer

✦ Table of Contents

Cover
CALCULUS with analytic geometry
Copyright
Preface v
To the Reader viii
Contents
1 Functions / 1
1.1 The Real Numbers / 2
1.2 Points and Lines in the Plane / 10
1.3 Functions / 23
1.4 Graphs / 31
1.5 Aids to Graphing / 39
1.6 Combining Functions / 47
1.7 Trigonometric Functions / 54
2 Limits and Continuity / 69
2.1 Definition of Limit / 70
2.2 Examples of Limits / 78
2.3 Basic Limit Theorems / 86
2.4 One-Sided Limits and Limits at Infinity / 105
2.5 Continuity / 121
2.6 Two Special Theorems on Continuous Functions / 128
3 Derivatives / 139
3.1 Tangent Lines / 140
3.2 The Derivative / 150
3.3 Combinations of Derivatives / 163
3.4 The Chain Rule / 177
3.5 Higher Derivatives / 186
3.6 Implicit Differentiation / 190
3.7 Related Rates / 195
3.8 Tangent Line Approximations and the Differential / 199
4 Applications of the Derivative / 207
4.1 Extreme Values of Differentiable Functions / 208
4.2 The Mean Value Theorem / 216
4.3 Increasing and Decreasing Functions / 221
4.4 The Second Derivative Test / 233
4.5 Concavity and Inflection Points / 239
4.6 Graphing / 247
5 The Integral / 257
5.1 Preparation for the Definite Integral / 259
5.2 The Definite Integral / 269
5.3 Special Properties of the Definite Integral / 276
5.4 The Fundamental Theorem of Calculus / 284
5.5 Indefinite Integrals and Integration Rules / 295
5.6 The Logarithm as an Integral / 302
5.7 Another Look at Area / 308
5.8 Who Invented Calculus? / 317
6 Techniques of Integration / 321
6.1 Integration by Parts / 322
6.2 Integration by Substitution / 329
6.3 Trigonometric Integrals / 337
6.4 Integration by Trigonometric Substitution / 346
6.5 Riemann Sums and the Riemann Integral / 353
6.6 The Trapezoidal Rule and Simpson's Rule / 363
6.7 Improper Integrals / 373
7 Applications of the Integral / 385
7.1 Volumes: The Cross-Sectional Method / 386
7.2 Volumes: The Shell Method / 394
7.3 Arc Length / 399
7.4 Work / 404
7.5 Moments and Centers of Gravity / 410
7.6 Hydrostatic Force / 421
7.7 Polar Coordinates / 424
7.8 Area in Polar Coordinates / 432
8 Inverse Functions / 441
8.1 Inverse Functions / 442
8.2 Continuity and Derivatives of Inverse Functions / 449
8.3 The Inverse Trigonometric Functions / 454
8.4 Exponential and Logarithmic Functions / 467
8.5 Exponential Growth and Decay / 481
8.6 Hyperbolic Functions / 487
8.7 Partial Fractions / 493
8.8 I'Hopital's Rule / 503
9 Sequences and Series / 517
9.1 Polynomial Approximation and Taylor's Theorem / 518
9.2 Sequences / 527
9.3 Infinite Series / 544
9.4 Nonnegative Series: The Integral Test and the Comparison Tests / 558
9.5 Nonnegative Series: The Ratio Test and the Root Test / 565
9.6 Alternating Series and Absolute Convergence / 569
9.7 Power Scries / 579
9.8 Taylor Scries / 595
9.9 Binomial Series / 604
10 Conic Sections / 613
10.1 The Parabola / 615
10.2 The Ellipse / 622
10.3 The Hyperbola / 632
10.4 Rotation of Axes / 639
10.5 A Unified Description of Conic Sections / 644
11 Vectors, Lines, and Planes / 655
11.1 Vectors in Space / 656
11.2 The Dot Product / 672
11.3 The Cross Product and Triple Products / 680 680
11.4 Lines in Space / 686
11.5 Planes in Space / 691
12 Vector-Valued Functions / 701
12.1 Definitions and Examples / 702
12.2 Limits and Continuity of Vector-Valued Functions / 710
12.3 Derivatives and Integrals of Vector-Valued Functions / 715
12.4 Space Curves and Their Lengths / 729
12.5 Tangents and Normals to Curves / 740
12.6 Kepler's Laws of Motion / 755
13 Partial Derivatives / 765
13.1 Functions of Several Variables / 766
13.2 Limits and Continuity / 778
13.3 Partial Derivatives / 787
13.4 The Chain Rule / 800
13.5 Directional Derivatives / 809
13.6 The Gradient / 813
13.7 Tangent Plane Approximations and Differentials / 820
13.8 Extreme Values / 824
13.9 Lagrange Multipliers / 834
14 Multiple Integrals / 847
14.1 Double Integrals / 848
14.2 Double Integrals in Polar Coordinates / 864
14.3 Surface Area / 873
14.4 Triple Integrals / 880
14.5 Triple Integrals in Cylindrical Coordinates / 893
14.6 Triple Integrals in Spherical Coordinates / 901
14.7 Moments and Centers of Gravity / 907
15 Calculus of Vector Fields / 917
15.1 Vector Fields / 918
15.2 Line Integrals / 930
15.3 The Fundamental Theorem of Line Integrals / 943
15.4 Green's Theorem / 949
15.5 Surface Integrals / 959
15.6 Stokes's Theorem / 977
15.7 The Divergence Theorem / 985
Appendixes
Appendix A Proofs of Selected Theorems A-2
Appendix B Differential Equations A-12
Tables A-33
Table of Integrals A-38
Answers to Selected Exercises A-43
Index of Symbols A-73
Index A-74