Exponential Functions
3.1
OBJECTIVE
• Graph exponential functions.
• Differentiate exponential functions.
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3.1 Exponential Functions
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.
x
x
1
x
Examples: f  x   2 , f  x     , f  x    0.4 
2
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3.1 Exponential Functions
Example 1: Graph f (x)  2 x. First, we find some function values.
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3.1 Exponential Functions
Quick Check 1
x
For f  x   3x , complete this table of function values. Graph f  x   3 .
1
3
9
27
1
3
1
9
1
27
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3.1 Exponential Functions
DEFINITION:
e  lim 1 h   2.718281828459
1h
h0
We call e the natural base.
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3.1 Exponential Functions
THEOREM 1
The derivative of the function f given by f  x   e x is
itself:
f  x   f  x ,
or
d x
x
e e
dx
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3.1 Exponential Functions
Example 2: Find dy/dx:
a) y  3e x ;
b) y  x 2 e x ;
dy
x
a)
3e
dx
d x
 3 e
dx
 3e x
 
d 2 x
b)
xe
dx

ex
c) y  3 .
x


x2  e x  e x  2 x

e x x2  2 x

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
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3.1 Exponential Functions
Example 2 (concluded):
d  ex 
c)

3 
dx  x 



x3  e x  e x  3 x 2
x 
3
2
x 2e x  x  3
x6
e x ( x  3)
x4
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3.1 Exponential Functions
Quick Check 2
Differentiate:
a.) y  6e ,
x
dy
x
x
6
e

6
e
 
dx
dy 3 x
x e   x3e x  e x 3x 2  x 2 e x ( x  3)
b.) y  x e ,

dx
3 x
xe x ( x  2)
ex
dy  e x  x 2 e x  e x (2 x)


c.) y  2 ,

2 
4
x4
x
dx  x 
x
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
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ex  x  2
x3
3.1 Exponential Functions
THEOREM 2
d f (x)
e
 e f ( x )  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.
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3.1 Exponential Functions
Example 3: Differentiate each of the following with
respect to x:
a) y  e8 x ;
b) y  e
d 8x
a)
e
dx
 x 2 4 x7
;
c) y  e
x 2 3
.
8x
e
8


8e8 x
d  x2  4 x 7 
b)
e
dx
e
 x2  4 x 7
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  2 x  4 
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3.1 Exponential Functions
Example 3 (concluded):
d
c)
e
dx
2
x 3
d  x 2 3 
e
dx

 e

1
2
 x 3  1
2


2
1
2
x2  3
xe


1
2
 2x
x 2 3
x2  3
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3.1 Exponential Functions
Quick Check 3
Differentiate:
a.) f  x   e4 x , f   x   e4 x  4   4e4 x
b.) g  x   e
x3 8 x
c.) h  x   e
x 5
2
, g   x   e x 8 x (3x 2  8)
3
1

1 2
, h  x    x  5 2  2 x  e
2
h '( x) 
xe
x 2 5
x2 5
x2  5
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3.1 Exponential Functions
Example 4: Graph h(x)  1 e2 x with x ≥ 0.
Analyze the graph using calculus.
First, we find some values, plot the points, and sketch
the graph.
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3.1 Exponential Functions
Example 4 (continued):
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3.1 Exponential Functions
Example 4 (continued):
a) Derivatives. Since h(x)  1 e2 x ,
h(x)  2e2 x
and
h(x)  4e2 x .
1
b) Critical values. Since e  2 x  0, the derivative
e
2 x
h(x)  2e  0 for all real numbers x. Thus, the
derivative exists for all real numbers, and the equation
h (x) = 0 has no solution. There are no critical values.
2 x
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3.1 Exponential Functions
Example 4 (continued):
2 x
h
(x)

2e
 0 for

c) Increasing. Since the derivative
all real numbers x, we know that h is increasing over
the entire real number line.
2 x

d) Inflection Points. Since h ( x)  4e  0 we
know that the equation h (x) = 0 has no solution.
Thus there are no points of inflection.
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3.1 Exponential Functions
Example 4 (concluded):
2 x

h
(
x
)


4
e
 0 for all real
e) Concavity. Since
numbers x, we know that h is decreasing and the
graph is concave down over the entire real number
line.
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3.1 Exponential Functions
Section Summary
• The exponential function f  x   e x, where e  2.71828 has the
derivative f   x   e x . That is, the slope of a tangent line to the graph
of y  e x is the same as the function value at x.
• The graph of f  x   e x is an increasing function with no critical
values, no maximum or minimum values, and no points of inflection.
The graph is concave up, with
lim f  x   
x
and
lim f  x   0
x
• Calculus is rich in applications of exponential functions.
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