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 and Python files
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 3. Exercises 64
4. Finding sign changes of a function 65
5. Increasing and decreasing functions 66
Chapter 1. Numbers and Functions 5 6. Examples 67
1. What is a number? 5 7. Maxima and Minima 69
2. Exercises 7 8. Must there always be a maximum? 71
3. Functions 8 9. Examples – functions with and without maxima or
4. Inverse functions and Implicit functions 10 minima 71
5. Exercises 13 10. General method for sketching the graph of a
function 72
Chapter 2. Derivatives (1) 15 11. Convexity, Concavity and the Second Derivative 74
1. The tangent to a curve 15 12. Proofs of some of the theorems 75
2. An example – tangent to a parabola 16 13. Exercises 76
3. Instantaneous velocity 17 14. Optimization Problems 77
4. Rates of change 17 15. Exercises 78
5. Examples of rates of change 18
Chapter 6. Exponentials and Logarithms (naturally) 81
6. Exercises 18
1. Exponents 81
2. Logarithms 82
Chapter 3. Limits and Continuous Functions 21
3. Properties of logarithms 83
1. Informal definition of limits 21
4. Graphs of exponential functions and logarithms 83
2. The formal, authoritative, definition of limit 22
5. The derivative of ax and the definition of e 84
3. Exercises 25
6. Derivatives of Logarithms 85
4. Variations on the limit theme 25
7. Limits involving exponentials and logarithms 86
5. Properties of the Limit 27
8. Exponential growth and decay 86
6. Examples of limit computations 27
9. Exercises 87
7. When limits fail to exist 29
8. What’s in a name? 32 Chapter 7. The Integral 91
9. Limits and Inequalities 33 1. Area under a Graph 91
10. Continuity 34 2. When f changes its sign 92
11. Substitution in Limits 35 3. The Fundamental Theorem of Calculus 93
12. Exercises 36 4. Exercises 94
13. Two Limits in Trigonometry 36 5. The indefinite integral 95
14. Exercises 38 6. Properties of the Integral 97
7. The definite integral as a function of its integration
Chapter 4. Derivatives (2) 41
bounds 98
1. Derivatives Defined 41
8. Method of substitution 99
2. Direct computation of derivatives 42
9. Exercises 100
3. Differentiable implies Continuous 43
4. Some non-differentiable functions 43 Chapter 8. Applications of the integral 105
5. Exercises 44 1. Areas between graphs 105
6. The Differentiation Rules 45 2. Exercises 106
7. Differentiating powers of functions 48 3. Cavalieri’s principle and volumes of solids 106
8. Exercises 49 4. Examples of volumes of solids of revolution 109
9. Higher Derivatives 50 5. Volumes by cylindrical shells 111
10. Exercises 51 6. Exercises 113
11. Differentiating Trigonometric functions 51 7. Distance from velocity, velocity from acceleration 113
12. Exercises 52 8. The length of a curve 116
13. The Chain Rule 52 9. Examples of length computations 117
14. Exercises 57 10. Exercises 118
15. Implicit differentiation 58 11. Work done by a force 118
16. Exercises 60 12. Work done by an electric current 119
Chapter 5. Graph Sketching and Max-Min Problems 63 Chapter 9. Answers and Hints 121
1. Tangent and Normal lines to a graph 63
2. The Intermediate Value Theorem 63 GNU Free Documentation License 125
3
, 1. APPLICABILITY AND DEFINITIONS 125
2. VERBATIM COPYING 125
3. COPYING IN QUANTITY 125
4. MODIFICATIONS 125
5. COMBINING DOCUMENTS 126
6. COLLECTIONS OF DOCUMENTS 126
7. AGGREGATION WITH INDEPENDENT WORKS 126
8. TRANSLATION 126
9. TERMINATION 126
10. FUTURE REVISIONS OF THIS LICENSE 126
11. RELICENSING 126
4
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 and Python files
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 3. Exercises 64
4. Finding sign changes of a function 65
5. Increasing and decreasing functions 66
Chapter 1. Numbers and Functions 5 6. Examples 67
1. What is a number? 5 7. Maxima and Minima 69
2. Exercises 7 8. Must there always be a maximum? 71
3. Functions 8 9. Examples – functions with and without maxima or
4. Inverse functions and Implicit functions 10 minima 71
5. Exercises 13 10. General method for sketching the graph of a
function 72
Chapter 2. Derivatives (1) 15 11. Convexity, Concavity and the Second Derivative 74
1. The tangent to a curve 15 12. Proofs of some of the theorems 75
2. An example – tangent to a parabola 16 13. Exercises 76
3. Instantaneous velocity 17 14. Optimization Problems 77
4. Rates of change 17 15. Exercises 78
5. Examples of rates of change 18
Chapter 6. Exponentials and Logarithms (naturally) 81
6. Exercises 18
1. Exponents 81
2. Logarithms 82
Chapter 3. Limits and Continuous Functions 21
3. Properties of logarithms 83
1. Informal definition of limits 21
4. Graphs of exponential functions and logarithms 83
2. The formal, authoritative, definition of limit 22
5. The derivative of ax and the definition of e 84
3. Exercises 25
6. Derivatives of Logarithms 85
4. Variations on the limit theme 25
7. Limits involving exponentials and logarithms 86
5. Properties of the Limit 27
8. Exponential growth and decay 86
6. Examples of limit computations 27
9. Exercises 87
7. When limits fail to exist 29
8. What’s in a name? 32 Chapter 7. The Integral 91
9. Limits and Inequalities 33 1. Area under a Graph 91
10. Continuity 34 2. When f changes its sign 92
11. Substitution in Limits 35 3. The Fundamental Theorem of Calculus 93
12. Exercises 36 4. Exercises 94
13. Two Limits in Trigonometry 36 5. The indefinite integral 95
14. Exercises 38 6. Properties of the Integral 97
7. The definite integral as a function of its integration
Chapter 4. Derivatives (2) 41
bounds 98
1. Derivatives Defined 41
8. Method of substitution 99
2. Direct computation of derivatives 42
9. Exercises 100
3. Differentiable implies Continuous 43
4. Some non-differentiable functions 43 Chapter 8. Applications of the integral 105
5. Exercises 44 1. Areas between graphs 105
6. The Differentiation Rules 45 2. Exercises 106
7. Differentiating powers of functions 48 3. Cavalieri’s principle and volumes of solids 106
8. Exercises 49 4. Examples of volumes of solids of revolution 109
9. Higher Derivatives 50 5. Volumes by cylindrical shells 111
10. Exercises 51 6. Exercises 113
11. Differentiating Trigonometric functions 51 7. Distance from velocity, velocity from acceleration 113
12. Exercises 52 8. The length of a curve 116
13. The Chain Rule 52 9. Examples of length computations 117
14. Exercises 57 10. Exercises 118
15. Implicit differentiation 58 11. Work done by a force 118
16. Exercises 60 12. Work done by an electric current 119
Chapter 5. Graph Sketching and Max-Min Problems 63 Chapter 9. Answers and Hints 121
1. Tangent and Normal lines to a graph 63
2. The Intermediate Value Theorem 63 GNU Free Documentation License 125
3
, 1. APPLICABILITY AND DEFINITIONS 125
2. VERBATIM COPYING 125
3. COPYING IN QUANTITY 125
4. MODIFICATIONS 125
5. COMBINING DOCUMENTS 126
6. COLLECTIONS OF DOCUMENTS 126
7. AGGREGATION WITH INDEPENDENT WORKS 126
8. TRANSLATION 126
9. TERMINATION 126
10. FUTURE REVISIONS OF THIS LICENSE 126
11. RELICENSING 126
4
