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CHAPTER
4
4.1
4.2
4.3
4.4
CHAPTER
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
CHAPTER
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
CHAPTER 7
7.1
7.2
7.3
7.4
7.5
CHAPTER 8
8.1
8.2
8.3
8.4
8.5
Contents
The
Chain
Rule
Derivatives by the Chain Rule
Implicit Differentiation and Related Rates
Inverse Functions and Their Derivatives
Inverses of Trigonometric Functions
Integrals
The Idea of the Integral
177
Antiderivatives
182
Summation vs. Integration
187
Indefinite Integrals and Substitutions
195
The Definite Integral
201
Properties of the Integral and the Average Value
206
The Fundamental Theorem and Its Consequences
213
Numerical Integration
220
Exponentials
and
Logarithms
An Overview
228
The Exponential
ex
236
Growth and Decay in Science and Economics
242
252
Separable Equations Including the Logistic Equation
259
Powers Instead of Exponentials
267
Hyperbolic Functions
277
Techniques
of Integration
Integration by Parts
Trigonometric Integrals
Trigonometric Substitutions
Partial Fractions
Improper Integrals
Applications
of
the
Integral
Areas and Volumes by Slices
Length of a Plane Curve
Area of a Surface of Revolution
Probability and Calculus
Masses and Moments
8.6
Force, Work, and Energy
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View Full DocumentCHAPTER
6
Exponentials and Logarithms
This chapter is devoted to exponentials like 2" and 10" and above all
ex.
The goal is
to understand them, differentiate them, integrate them, solve equations with them,
and invert them (to reach the logarithm). The overwhelming importance of ex makes
this a crucial chapter in pure and applied mathematics.
In the traditional order of calculus books, ex waits until other applications of the
.
integral are complete. I would like to explain why it is placed earlier here. I believe
that the equation
dyldx
=
y
has to be emphasized above techniques of integration.
The laws of nature are expressed by
drflerential equations,
and at the center is
ex.
Its
applications are to life sciences and physical sciences and economics and engineering
(and morewherever
change is influenced by the present state). The model produces
a differential equation and I want to show what calculus can do.
The key is always
bm+"
=
(bm)(b3. Section 6.1 applies that rule in three ways:
1. to understand the
logarithm
as the
exponent;
2. to draw
graphs
on ordinary and semilog and loglog paper;
3. to find
derivatives.
The slope of b" will use bX+*"
=
(bx)(bh").
h
6.1
An
Overview
There is a good chance you have met logarithms. They turn multiplication into
addition, which is a lot simpler. They are the basis for slide rules (not so important)
and for graphs on log paper (very important). Logarithms are mirror images of
exponentialsand
those I know you have met.
Start with exponentials. The numbers 10 and lo2and lo3 are basic to the decimal
system. For completeness I also include lo0, which is "ten to the zeroth power" or
1.
The logarithms of those numbers are the exponents.
The logarithms of 1 and 10 and
100 and 1000 are 0 and 1 and 2 and 3. These are logarithms "to base
10,"
because
the powers are powers of 10.
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 Fall '08
 PETRINA

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