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[PDF] MATH 221 FIRST SEMESTER CALCULUS

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217265[PDF] MATH 221 FIRST SEMESTER CALCULUS

MATH 221

FIRST SEMESTER

CALCULUS

fall 2009

Typeset:June 8, 2010

1

MATH 221 { 1st SEMESTER CALCULUS

LECTURE NOTES VERSION 2.0 (fall 2009)This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting

from an extensive collection of notes and problems compiled by Joel Robbin. The LATEX andPythonles which were used to produce these notes are available at the following web site http://www.math.wisc.edu/ ~angenent/Free-Lecture-Notes They are meant to be freely available in the sense that \free software" is free. More precisely: Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

1.2 or any later version published by the Free Software Foundation; with no Invariant

Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Contents

Chapter 1. Numbers and Functions

5

1. What is a number?

5

2. Exercises

7

3. Functions

8

4. Inverse functions and Implicit functions

10

5. Exercises

13

Chapter 2. Derivatives (1)

15

1. The tangent to a curve

15

2. An example { tangent to a parabola

16

3. Instantaneous velocity

17

4. Rates of change

17

5. Examples of rates of change

18

6. Exercises

18

Chapter 3. Limits and Continuous Functions

21

1. Informal denition of limits

21

2. The formal, authoritative, denition of limit

22

3. Exercises

25

4. Variations on the limit theme

25

5. Properties of the Limit

27

6. Examples of limit computations

27

7. When limits fail to exist

29

8. What's in a name?

32

9. Limits and Inequalities

33

10. Continuity

34

11. Substitution in Limits

35

12. Exercises

36

13. Two Limits in Trigonometry

36

14. Exercises

38

Chapter 4. Derivatives (2)

41

1. Derivatives Dened

41

2. Direct computation of derivatives

42

3. Dierentiable implies Continuous

43

4. Some non-dierentiable functions

43

5. Exercises

44

6. The Dierentiation Rules

45

7. Dierentiating powers of functions

48

8. Exercises

49

9. Higher Derivatives

50

10. Exercises

51

11. Dierentiating Trigonometric functions

51

12. Exercises

52

13. The Chain Rule

52

14. Exercises

57

15. Implicit dierentiation

58

16. Exercises

60

Chapter 5. Graph Sketching and Max-Min Problems

63

1. Tangent and Normal lines to a graph

63

2. The Intermediate Value Theorem

63 3. Exercises64

4. Finding sign changes of a function

65

5. Increasing and decreasing functions

66

6. Examples

67

7. Maxima and Minima

69

8. Must there always be a maximum?

71 9. Examples { functions with and without maxima or

minima 71

10. General method for sketching the graph of a

function 72

11. Convexity, Concavity and the Second Derivative

74

12. Proofs of some of the theorems

75

13. Exercises

76

14. Optimization Problems

7 7

15. Exercises

78
Chapter 6. Exponentials and Logarithms (naturally) 81

1. Exponents

81

2. Logarithms

82

3. Properties of logarithms

83

4. Graphs of exponential functions and logarithms

83

5. The derivative ofaxand the denition ofe84

6. Derivatives of Logarithms

85

7. Limits involving exponentials and logarithms

86

8. Exponential growth and decay

86

9. Exercises

87

Chapter 7. The Integral

91

1. Area under a Graph

91

2. Whenfchanges its sign92

3. The Fundamental Theorem of Calculus

93

4. Exercises

94

5. The indenite integral

95

MATH 221

FIRST SEMESTER

CALCULUS

fall 2009

Typeset:June 8, 2010

1

MATH 221 { 1st SEMESTER CALCULUS

LECTURE NOTES VERSION 2.0 (fall 2009)This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting

from an extensive collection of notes and problems compiled by Joel Robbin. The LATEX andPythonles which were used to produce these notes are available at the following web site http://www.math.wisc.edu/ ~angenent/Free-Lecture-Notes They are meant to be freely available in the sense that \free software" is free. More precisely: Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

1.2 or any later version published by the Free Software Foundation; with no Invariant

Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Contents

Chapter 1. Numbers and Functions

5

1. What is a number?

5

2. Exercises

7

3. Functions

8

4. Inverse functions and Implicit functions

10

5. Exercises

13

Chapter 2. Derivatives (1)

15

1. The tangent to a curve

15

2. An example { tangent to a parabola

16

3. Instantaneous velocity

17

4. Rates of change

17

5. Examples of rates of change

18

6. Exercises

18

Chapter 3. Limits and Continuous Functions

21

1. Informal denition of limits

21

2. The formal, authoritative, denition of limit

22

3. Exercises

25

4. Variations on the limit theme

25

5. Properties of the Limit

27

6. Examples of limit computations

27

7. When limits fail to exist

29

8. What's in a name?

32

9. Limits and Inequalities

33

10. Continuity

34

11. Substitution in Limits

35

12. Exercises

36

13. Two Limits in Trigonometry

36

14. Exercises

38

Chapter 4. Derivatives (2)

41

1. Derivatives Dened

41

2. Direct computation of derivatives

42

3. Dierentiable implies Continuous

43

4. Some non-dierentiable functions

43

5. Exercises

44

6. The Dierentiation Rules

45

7. Dierentiating powers of functions

48

8. Exercises

49

9. Higher Derivatives

50

10. Exercises

51

11. Dierentiating Trigonometric functions

51

12. Exercises

52

13. The Chain Rule

52

14. Exercises

57

15. Implicit dierentiation

58

16. Exercises

60

Chapter 5. Graph Sketching and Max-Min Problems

63

1. Tangent and Normal lines to a graph

63

2. The Intermediate Value Theorem

63 3. Exercises64

4. Finding sign changes of a function

65

5. Increasing and decreasing functions

66

6. Examples

67

7. Maxima and Minima

69

8. Must there always be a maximum?

71 9. Examples { functions with and without maxima or

minima 71

10. General method for sketching the graph of a

function 72

11. Convexity, Concavity and the Second Derivative

74

12. Proofs of some of the theorems

75

13. Exercises

76

14. Optimization Problems

7 7

15. Exercises

78
Chapter 6. Exponentials and Logarithms (naturally) 81

1. Exponents

81

2. Logarithms

82

3. Properties of logarithms

83

4. Graphs of exponential functions and logarithms

83

5. The derivative ofaxand the denition ofe84

6. Derivatives of Logarithms

85

7. Limits involving exponentials and logarithms

86

8. Exponential growth and decay

86

9. Exercises

87

Chapter 7. The Integral

91

1. Area under a Graph

91

2. Whenfchanges its sign92

3. The Fundamental Theorem of Calculus

93

4. Exercises

94

5. The indenite integral

95

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