Calculus with analytic geometry

Description

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 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