Math 221 – 1st Semester Calculus - Department of Mathematics

Math 221 ? 1st Semester Calculus

Lecture Notes for Fall 2006. Prof J. Robbin

December 21, 2006

All references to "Thomas" or "the textbook" in these notes refer to

Thomas' CALCULUS 11th edition

published by Pearson Addison Wesley in 2005. These notes may be downloaded from

. Some portions of these notes are adapted from

. Some problems come from a list compiled by Arun Ram, others come from the WES program, and others come from the aforementioned Thomas text or from Stewart, Calculus (Early Transcendentals), 3rd Edition.

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Contents

I What you need to know to take Calculus 221

5

1 Algebra

5

2 Coordinate Geometry

7

3 Functions

9

4 Trigonometry

12

5 Additional Exercises

15

II Limits

17

6 Tangent and Velocity

17

7 Limits

19

8 Two Limits in Trigonometry

26

9 Continuity

27

III Differentiation

31

10 Derivatives Defined

31

11 Higher Derivatives and Differential Notation

37

12 Implicit Functions

41

13 The Chain Rule

43

14 Inverse Functions

47

15 Differentiating Trig Functions

50

16 Exponentials and Logarithms

53

17 Parametric Equations

59

18 Approximation*

63

19 Additional Exercises

67

2

IV Applications of Derivatives

68

20 The Derivative as A Rate of Change

68

21 Related Rates

71

22 Some Theorems about Derivatives

74

23 Curve Plotting

81

24 Max Min Word Problems

84

25 Exponential Growth

87

26 Indeterminate Forms (l'H^opital's Rule)

91

27 Antiderivatives

93

28 Additional Problems

95

V Integration

96

29 The Definite Integral

96

30 The Fundamental Theorem of Calculus

104

31 Averages

107

32 Change of Variables

108

VI Applications of Definite Integrals

112

33 Plane Area

112

34 Volumes

114

35 Arc Length

118

36 Surface Area

122

37 Center of Mass

125

VII Loose Ends

132

38 The Natural Log Again*

132

3

39 Taylor Approximation*

133

40 Newton's Method*

135

VIII More Problems

136

IX Notes for TA's

153

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Chapter I

What you need to know to take Calculus 221

In this chapter we will review material from high school mathematics. We also teach some of this material at UW in Math 112, Math 113, and Math 114, and some of it will be reviewed again as we need it. This material should be familiar to you. If it is not, you may not be ready for calculus. Pay special attention to the definitions. Important terms are shown in boldface when they are first defined.

1 Algebra

This section contains some things which should be easy for you. (If they are not, you may not be ready for calculus.)

?1.1. Answer these questions. 1. Factor x2 - 6x + 8.

2. Find the values of x which satisfy x2 - 7x + 9 = 0. (Quadratic formula.)

3. x2 - y2 =? Does x2 + y2 factor?

4. True or False: x2 + 4 = x + 2?

5.

True or False:

(9x)1/2

= 3 x?

6.

True or False:

x2x8 x3

= x2+8-3 = x7?

7. Find x if 3 = log2(x). 8. What is log7(7x)?

9. True or False: log(x + y) = log(x) + log(y)?

10. True or False: sin(x + y) = sin(x) + sin(y)?

?1.2. There are conventions about the order of operations. For example,

ab + c means (ab) + c and not a(b + c),

a

b c a b

c

log a + b

means a/(b/c) and not (a/b)/c,

means (a/b)/c and not a/(b/c), means (log a) + b and not log(a + b).

If necessary, we use parentheses to indicate the order of doing the operations.

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