Chapter 3.01
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Transcript Chapter 3.01
4.4
Exponential Functions
OBJECTIVES
Differentiate exponential functions.
Solve application problems with
exponential functions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION:
An exponential function f is given by
f (x) a x ,
where x is any real number, a > 0, and a ≠ 1. The
number a is called the base.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.1 - 2
Example 1: Look at the graph
function values.
First, we find some
f (x) 2 x.
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Slide 3.1 - 3
DEFINITION:
e is a number, named for the Swiss mathematician
Leonhard Euler.
e lim 1 h 2.718281828459
1h
h0
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Slide 3.1 - 4
THEOREM 1
d x
e ex
dx
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Slide 3.1 - 5
Example 1: Find dy/dx:
a) y 3e ;
x
b) y x e ;
2 x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
x
e
c) y 3 .
x
Slide 3.1 - 6
dy
x
a)
3e
dx
d 2 x
b)
x e
dx
d x
3 e
dx
x
3e
x e e 2x
e x 2x
2
x
x
x
2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.1 - 7
Example 1 (concluded):
x
d e
c)
dx x 3
x e e 3x
3
x
x
2
x
3 2
x e x 3
6
x
e x (x 3)
4
x
2 x
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Slide 3.1 - 8
THEOREM 2
d f (x)
f (x)
e e f (x)
dx
OR
d u
u du
e e
dx
dx
The derivative of e to some power is the product of e
to that power and the derivative of the power.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.1 - 9
Example 2: Differentiate each of the following with
respect to x:
a) y e ;
b) y e
8x
c) y e
x 2 3
x 2 4 x 7
;
.
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Slide 3.1 - 10
d 8x
a)
e
dx
e 8
8x
d x 2 4 x7
b)
e
dx
8e
8x
e
x 2 4 x7
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2x 4
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Example 2 (concluded):
d
c)
e
dx
x 3
2
d x 2 3
e
dx
1
2
1
2
x 3
1 2
e
x 3 2 x
2
1
2
2
x 2 3
xe
x 3
2
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Slide 3.1 - 12
The sales of a new computer ( in thousands) are
given by: S (t ) 100 90e0.3t
t represents time in years.
Find the rate of change of sales at each time.
a) after 1 year
b) after 5 years
c) What is happening to the rate of change of sales?
Answers: a) 20 b) 6 c) decreasing
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Slide 3.1 - 13
Write an equation of the tangent line to
f ( x) 2e
3 x
at x = 0.
Answer: y = -6x+2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.1 - 14