Calculo

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CONTENTS
Introduction

.............................................................. 1

Chapter 1.

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 2.

Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 3.

The Derivative . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 4.

Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 99

Chapter 5.

Analysis of Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . 139

Chapter 6.

Applications of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 177

Chapter 7.

Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Chapter 8.

Applications of the Definite Integral
in Geometry, Science, and Engineering . . . . . . . . . . . . . . . . . . . . . . . . . 256

Chapter 9.

Principles of Integral Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 292

Chapter 10.

Mathematical Modeling with Differential Equations . . . . . . . . . . . . . . 343

Chapter 11.

Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Chapter 12.

Analytic Geometry in Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .408

Chapter 13.

Three-DimensionalSpace; Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

Chapter 14.

Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

Chapter 15.

Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Chapter 16.

Multiple Integrals . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Chapter 17.

Topics in Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

Appendix A. Real Numbers, Intervals, and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 640
Appendix B. Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 647
Appendix C. Coordinate Planes and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
Appendix D. Distance, Circles, and Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . 658
Appendix E. Trigonometry Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
Appendix F.

Solving Polynomial Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

CALCULUS:

A New Horizon from Ancient Roots
EXERCISE SET FOR INTRODUCTION
1.

(a)

x = 0.123123123 . . .; 1000x = 123.123123123 . . . = 123 + x; 999x = 123; x =

(b)

x = 12.7777 . . .; 10x = 127.7777 . . ., so 9x = 10x − x = 115; x =

(c)

41
123
=
999
333

x = 38.07818181 . . .; 100x = 3807.818181 . ..; 99x = 3769.74;
376974
41886
20943
3769.74
=
=
=
x=
99
9900
1100
550
537
4296
=
0.4296000 . . . = 0.4296 =
10000
1250

(d)

115
9

repeats

2.

3.

22
= 3. 142857 . . .
(a) π is irrational, and thus has a nonrepeating decimal expansion, whereas
7
22
(b)

7


333
63 17 + 15 5
22
63 17 + 15 5
223
355


<
<
<
(b)
(a)
<
71
106
25 7 + 155
113
7
25 7 + 15 5

63 17 + 15 5
333

(d)
(c)
106
25 7 + 15 5
2

4.

16
8
D=
r
9
9
πr2 so 256/81 was the approximation used for π .

2

=

256 2
r . The area of a circle of radius r is
81

6.

7.

If r is the radius, then D = 2r so

(b)
5.

(a)

256/81 ≈ 3.16049, 22/7 ≈ 3.14268, and π ≈ 3.14159 so 256/81 is worse than 22/7.

The first series, taken...
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