Preface to the e-text edition |
vii |
|
1 |
Functions and Their Properties |
1 |
1.1 |
The Meaning of a Function |
1 |
1.2 |
Function Values and the Box Method |
5 |
1.3 |
The Absolute Value of a Function |
12 |
1.4 |
A Quick Review of Inequalities |
21 |
1.4 .1 |
The triangle inequalities |
23 |
1.5 |
Chapter Exercises |
30 |
1.6 |
Using Computer Algebra Systems (CAS), |
31 |
2 |
Limits and Continuity |
33 |
2.1 |
One-Sided Limits of Functions |
35 |
2.2 |
Two-Sided Limits and Continuity |
40 |
2.3 |
Important Theorems About Continuous Functions |
59 |
2.4 |
Evaluating Limits at Infinity |
63 |
2.5 |
How to Guess a Limit |
66 |
2.6 |
Chapter Exercises |
76 |
3 |
The Derivative of a Function |
79 |
3.1 |
Motivation |
80 |
3.2 |
Working with Derivatives |
89 |
3.3 |
The Chain Rule |
95 |
3.4 |
Implicit Functions and Their Derivatives |
108 |
3.5 |
Derivatives of Trigonometric Functions |
113 |
3.6 |
Important Results About Derivatives |
121 |
3.7 |
Inverse Functions |
128 |
3.8 |
Inverse Trigonometric Functions |
136 |
3.9 |
Derivatives of Inverse Trigonometric Functions |
140 |
3.1.0 |
Relating Rates of Change |
143 |
3.1.1 |
Newton’s Method for Calculating Roots |
151 |
3.1.2 |
L’Hospital’s Rule |
161 |
3.1.3 |
Chapter Exercises |
173 |
3.1.4 |
Challenge Questions |
174 |
3.1.5 |
Using Computer Algebra Systems |
175 |
4 |
Exponentials and Logarithms |
177 |
4.1 |
Exponential Functions and Their Logarithms |
178 |
4.2 |
Euler’s Number, e = 2.718281828 |
184 |
4.3 |
Euler’s Exponential Function and the Natural Logarithm |
189 |
4.4 |
Derivative of the Natural Logarithm |
193 |
4.5 |
Differentiation Formulae for General Exponential Functions . . . |
196 |
4.6 |
Differentiation Formulae for General Logarithmic Functions . . . |
201 |
4.7 |
Applications |
204 |
4.8 |
Chapter Exercises |
210 |
4.9 |
Using Computer Algebra Systems |
211 |
5 |
Curve Sketching |
213 |
5.1 |
Solving Polynomial Inequalities |
213 |
5.2 |
Solving Rational Function Inequalities |
225 |
5.3 |
Graphing Techniques |
232 |
5.4 |
Application of Derivatives to Business and Economics |
256 |
5.5 |
Single variable optimization problems |
258 |
5.6 |
Chapter Exercises |
259 |
6 |
Integration |
261 |
6.1 |
Antiderivatives and the Indefinite Integral |
262 |
6.2 |
Definite Integrals |
277 |
6.3 |
The Summation Convention |
286 |
6.4 |
Area and the Riemann Integral |
291 |
6.5 |
Chapter Exercises |
304 |
6.6 |
Using Computer Algebra Systems |
306 |
7 |
Techniques of Integration |
309 |
7.1 |
Trigonometric Identities |
309 |
7.2 |
The Substitution Rule |
311 |
7.3 |
Integration by Parts |
323 |
7.3 .1 |
The Product of a Polynomial and a Sine or Cosine |
328 |
7.3 .2 |
The Product of a Polynomial and an Exponential |
331 |
7.3 .3 |
The Product of a Polynomial and a Logarithm |
334 |
7.3 .4 |
The Product of an Exponential and a Sine or Cosine . . . |
337 |
7.4 |
Partial Fractions |
346 |
7.4 .1 |
Review of Long Division of Polynomials |
347 |
7.4 .2 |
The Integration of Partial Fractions |
350 |
7.5 |
Products of Trigonometric Functions |
365 |
7.5 .1 |
Products of Sines and Cosines |
365 |
7.5 .2 |
Fourier Coefficients |
374 |
7.5 .3 |
Products of Secants and Tangents |
378 |
7.6 |
Trigonometric Substitutions |
386 |
7.6 .1 |
Completing the Square in a Quadratic (Review) |
386 |
7.6 .2 |
Trigonometric Substitutions |
390 |
7.7 |
Numerical Integration |
400 |
7.7 .1 |
The Trapezoidal Rule |
401 |
7.7 .2 |
Simpson’s Rule for n Even |
408 |
7.8 |
Improper Integrals |
414 |
7.9 |
Rationalizing Substitutions |
429 |
7.9 .1 |
Integrating rational functions of trigonometric expressions |
432 |
7.10 |
Chapter Exercises |
437 |
7.11 |
Using Computer Algebra Systems |
443 |
8 |
Applications of the Integral |
445 |
8.1 |
Motivation |
445 |
8.2 |
Finding the Area Between Two Curves |
448 |
8.3 |
The Volume of a Solid of Revolution |
464 |
8.4 |
Measuring the length of a curve |
477 |
8.5 |
Moments and Centers of Mass |
489 |
8.6 |
Chapter Exercises |
502 |
8.7 |
Using Computer Algebra Systems |
502 |
9 |
Simple Differential Equations |
505 |
9.1 |
Why Study Differential Equations? |
505 |
9.2 |
First-order Separable Equations |
512 |
9.3 |
Laws of Growth and Decay |
518 |
9.4 |
Using Computer Algebra Systems |
525 |
10 |
Multivariable Optimization Techniques |
527 |
10.1 |
Functions of More Than One Variable |
527 |
10.2 |
Continuity |
528 |
10.2 .1 |
Discontinuity at a point |
529 |
10.3 |
Partial Derivatives |
531 |
10.4 |
Higher Order Partial Derivatives |
533 |
10.5 |
The Chain Rule for Partial Derivatives |
535 |
10.6 |
Extrema of Functions of Two Variables |
540 |
10.6 .1 |
Maxima and Minima |
540 |
10.6 .2 |
The method of Lagrange multipliers |
544 |
10.7 |
Chapter Exercises |
551 |
11 |
Advanced Topics |
553 |
11.1 |
Infinite Sequences |
553 |
11.2 |
Sequences with Infinite Limits |
560 |
11.3 |
Limits from the Right |
563 |
11.4 |
Limits from the Left |
569 |
11.5 |
Summary |
575 |
11.6 |
Continuity |
576 |
11.7 |
Limits of Functions at Infinity |
578 |
11.8 |
Infinite Limits of Functions |
581 |
11.9 |
The Epsilon-Delta Method of Proof |
585 |
12 |
Appendix A: Review of Exponents and Radicals |
597 |
13 |
Appendix B: The Straight Line |
603 |
14 |
Appendix C: A Quick Review of Trigonometry |
609 |
14.1 |
The right-angled isosceles triangle (RT45) |
610 |
14.2 |
The RT30 triangle |
610 |
14.3 |
The basic trigonometric functions |
611 |
14.4 |
Identities |
613 |
14.4.1 |
The Law of Sines |
614 |
14.4.2 |
The Law of Cosines |
615 |
14.4.3 |
Identities for the sum and difference of angles |
616 |
15 |
Appendix D: The Natural Domain of a Function |
623 |
Solutions Manual |
627 |
|
1.1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
627 |
1.2 |
Exercise Set 1 (page 10) |
627 |
1.3 |
Exercise Set 2 (page 19) |
627 |
1.4 |
Exercise Set 3 (page 28) |
628 |
1.5 |
Chapter Exercises (page 30 ) |
629 |
Solutions |
|
631 |
2.1 |
Exercise Set 4 (page 39) |
631 |
2.2 |
Exercise Set 5 (page 45) |
631 |
2.2 |
Exercise Set 6 (page 49) |
631 |
2.2 |
Exercise Set 7 (page 56) |
632 |
2.2 |
Exercise Set 8 (page 57) |
632 |
2.3 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
633 |
2.4 |
Exercise Set 9 (page 65) |
633 |
2.5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
633 |
2.6 |
Chapter Exercises (page 76) |
633 |
Solutions |
|
635 |
3.1 |
Exercise Set 10 (page 88) |
635 |
3.2 |
Exercise Set 11 (page 93) |
635 |
3.3 |
Exercise Set 12 (page 105) |
636 |
3.4 |
Exercise Set 13 (page 112) |
637 |
3.5 |
Exercise Set 14 (page 119) |
637 |
3.6 |
Exercise Set 15 (page 127) |
637 |
3.7 |
Exercise Set 16 (page 134) |
639 |
3.8 |
Exercise Set 17 (page 139) |
639 |
3.9 |
Exercise Set 18 (page 142) |
639 |
3.1.0 |
Special Exercise Set (page 149) |
640 |
3.1.1 |
Exercise Set 19 (page 160) |
640 |
3.1.2 |
Exercise Set 20 (page 172) |
641 |
3.1.3 |
Chapter Exercises (page 173) |
641 |
Solutions |
|
643 |
4.1 |
Exercise Set 21 (page 183) |
643 |
4.2 |
Exercise Set 22 (page 188) |
643 |
4.3 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
644 |
4.4 |
Exercise Set 23 (page 196) |
644 |
4.5 |
Exercise Set 24 (page 199) |
644 |
4.6 |
Exercise Set 25 (page 203) |
645 |
4.7 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
646 |
4.8 |
Chapter Exercises (page 210) |
646 |
Solutions |
|
649 |
5.1 |
Exercise Set 26 (page 216) |
649 |
5.2 |
Exercise Set 27 (page 223) |
649 |
5.3 |
Exercise Set 28 (page 230) |
650 |
5.4 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
652 |
5.5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
652 |
5.6 |
Single variable optimization problems (page 258) |
653 |
5.7 |
Chapter Exercises: Use Plotter (page 259) |
654 |
Solutions |
|
657 |
6.1 |
Exercise Set 29 (page 275) |
657 |
6.2 |
Exercise Set 30 (page 289) |
657 |
6.3 |
Exercise Set 31 (page 284) |
659 |
6.4 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
660 |
6.5 |
Chapter Exercises (page 304) |
660 |
Solutions |
665 |
|
7.1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
665 |
7.2 |
Exercise Set 32 (page 321) |
665 |
7.3 |
Exercise Set 33 (page 345) |
666 |
7.4 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
667 |
7.4 .1 |
Exercise Set 34 (page 349) |
667 |
7.4 .2 |
Exercise Set 35 (page 364) |
668 |
7.5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
670 |
7.5 .1 |
Exercise Set 36 (page 373) |
670 |
7.5 .2 |
Exercise Set 37 (page 385) |
671 |
7.6 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
672 |
7.6 .1 |
Exercise Set 38 (page 390) |
672 |
7.6 .2 |
Exercise Set 39 (page 398) |
673 |
7.7 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
675 |
7.7 .1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
675 |
7.7 .2 |
Exercise Set 40 (page 411) |
675 |
7.8 |
Exercise Set 41 (page 426) |
677 |
7.9 |
Chapter Exercises (page 437) |
678 |
Solutions |
|
693 |
8.1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
693 |
8.2 |
Exercise Set 42 (page 452) |
693 |
8.2 .1 |
Exercise Set 43 (page 462) |
694 |
8.3 |
Exercise Set 44 (page 476) |
695 |
8.4 |
Exercise Set 45 (page 487) |
696 |
8.5 |
Exercise Set 46 (page 500) |
697 |
8.6 |
Chapter Exercises (page 502) |
698 |
Solutions |
|
701 |
9.1 |
Exercise Set 47 (page 511) |
701 |
9.2 |
Exercise Set 48 (page 518) |
701 |
9.3 |
Exercise Set 49 (page 523) |
702 |
Solutions |
|
705 |
10.1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
705 |
10.2 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
705 |
10.3 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
705 |
10.4 |
Exercise Set 50 (page 535) |
705 |
10.5 |
Exercise Set 51 (page 548) |
706 |
10.6 |
Chapter Exercises (page 551) |
707 |
Solutions |
|
711 |
11.1 |
Exercise Set 52 (page 559) |
711 |
11.2 |
Exercise Set 53 (page 563) |
711 |
11.3 |
Exercise Set 54 (page 574) |
712 |
11.4 |
Exercise Set 55 (page 580) |
712 |
11.5 |
Exercise Set 56 (page 584) |
712 |
11.6 |
Exercise Set 57 (page 594) |
713 |
Solutions to Problems in the Appendices |
|
715 |
1 |
APPENDIX A - Exercise Set 58 (page 601) |
715 |
12.2 |
APPENDIX B - Exercise Set 59 (page 607) |
716 |
12.3 |
APPENDIX C - Exercise Set 60 (page 620) |
716 |
12.4 |
APPENDIX D - Exercise Set 61 (page 626) |
718 |
Acknowledgments |
719 |
|
Credits |
721 |
|
About the Author |
723 |
|
Index |
724 |
|