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.
1
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
4
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.
5
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