Calculus: Analytic geometry and calculus, with vectors

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Table of Contents......Page 10
1.1 Real numbers......Page 16
1.2 Slopes and equations of lines......Page 25
1.3 Lines and linear equations; parallelism and perpendicularity......Page 30
1.4 Distances, circles, and parabolas......Page 39
1.5 Equations, statements, and graphs......Page 50
1.6 Introduction to velocity and acceleration......Page 56
2.1 Vectors in E^3......Page 63
2.2 Coordinate systems and vectors in E^3......Page 74
2.3 Scalar products, direction cosines, and lines in E^3......Page 82
2.4 Planes and lines in E^3......Page 93
2.5 Determinants and applications......Page 102
2.6 Vector products and changes of coordinates in E^3......Page 112
3.1 Functional notation......Page 126
3.2 Limits......Page 137
3.3 Unilateral limits and asymptotes......Page 148
3.4 Continuity......Page 159
3.5 Difference quotients and derivatives......Page 167
3.6 The chain rule and differentiation of elementary functions......Page 179
3.7 Rates, velocities......Page 191
3.8 Related rates......Page 203
3.9 Increments and differentials......Page 208
4.1 Indefinite integrals......Page 217
4.2 Riemann sums and integrals......Page 227
4.3 Properties of integrals......Page 239
4.4 Areas and integrals......Page 250
4.5 Volumes and integrals......Page 259
4.6 Riemann-Cauchy integrals and work......Page 265
4.7 Mass, linear density, and moments......Page 274
4.8 Moments and centroids in B2 and E3......Page 284
4.9 Simpson and other approximations to integrals......Page 291
5.1 Graphs, slopes, and tangents......Page 299
5.2 Trends, maxima, and minima......Page 309
5.3 Second derivatives, convexity, and flexpoints......Page 319
5.4 Theorems about continuous and differentiable functions......Page 328
5.5 The Rolle theorem and the mean-value theorem......Page 339
5.6 Sequences, series, and decimals......Page 349
5.7 Darboux sums and Riemann integrals......Page 359
6.1 Parabolas......Page 369
6.2 Geometry of cones and conics......Page 376
6.3 Ellipses......Page 384
6.4 Hyperbolas......Page 394
6.5 Translation and rotation of axes......Page 407
6.6 Quadric surfaces......Page 417
7.1 Curves and lengths......Page 423
7.2 Lengths and integrals......Page 432
7.3 Center and radius of curvature......Page 443
8.1 Trigonometric functions and their derivatives......Page 453
8.2 Trigonometric integrands......Page 464
8.3 Inverse trigonometric functions......Page 473
8.4 Integration by trigonometric and other substitutions......Page 484
8.5 Integration by substituting z = tan e/2......Page 492
9.1 Exponentials and logarithms......Page 495
9.2 Derivatives and integrals of exponentials and logarithms......Page 506
9.3 Hyperbolic functions......Page 520
9.4 Partial fractions......Page 526
9.5 Integration by parts......Page 533
10.1 Geometry of coordinate systems......Page 541
10.2 Polar curves, tangents, and lengths......Page 553
10.3 Areas and integrals involving polar coordinates......Page 562
11.1 Elementary partial derivatives......Page 568
11.2 Increments, chain rule, and gradients......Page 577
11.3 Formulas involving partial derivatives......Page 591
12.1 Definitions and basic theorems......Page 602
12.2 Ratio test and integral test......Page 614
12.3 Alternating series and Fourier series......Page 625
12.4 Power series......Page 634
12.5 Taylor formulas with remainders......Page 647
12.6 Euler-Maclaurin summation formulas......Page 655
13.1 Iterated integrals......Page 667
13.2 Iterated integrals and volumes......Page 674
13.3 Double integrals......Page 682
13.4 Rectangular coordinate applications of double and iterated integrals......Page 691
13.5 Integrals in polar coordinates......Page 702
13.6 Triple integrals; rectangular coordinates......Page 710
13.7 Triple integrals; cylindrical coordinates......Page 717
13.8 Triple integrals; spherical coordinates......Page 722
APPENDIX 1 Proofs of basic theorems on limits......Page 730
APPENDIX 2 Volumes......Page 736
INDEX......Page 740