Download 5.4 Exponential Functions: Differentiation and Integration

5.4
Differentiation and Integration of
“E”
2012
The Natural Exponential Function
The function f(x) = ln x is increasing on its entire domain,
and therefore it has an inverse function f –1.
The domain of f –1 is the set of all reals, and the range is the
set of positive reals, as shown in Figure 5.19.
Figure 5.19
The Natural Exponential Function
So, for any real number x,
If x happens to be rational, then
Because the natural logarithmic function is one-to-one, you
can conclude that f –1(x) and ex agree for rational values of x.
The Natural Exponential Function
The following definition extends the meaning of ex to
include all real values of x.
The inverse relationship between the natural logarithmic
function and the natural exponential function can be
summarized as follows.
5.4 Exponential Functions:
Differentiation and Integration
Definition of the Natural Exponential Function
The inverse of f ( x)  ln x is called the natural exponential
function and is denoted by
f ( x)  e .
-1
x
That is,
y  e  ln y  x.
x
Solve
for x
5.4 Exponential Functions:
Differentiation and Integration
This inverse relationship 
ln(e )  x and e
x
Solve 7  e
ln x
x
x 1
x 1
ln 7  ln(e )
ln 7  x  1
x  ln 7 1 
0.946
5.4 Exponential Functions:
Differentiation and Integration
Solve ln(2 x  3)  5
e  2x  3
5
2x  e  3
1 5
x   e  3
2
5
x  75.707
5.4 Exponential Functions:
Differentiation and Integration
THM 5.10 Operations with Exponential Functions
Let a and b be any real numbers.
1. e e  e
a b
a
a b
e
a b
2. b  e
e
5.4 Exponential Functions:
Differentiation and Integration
Use the Laws of Logarithms to prove: e e  e
a b
ln(e e )  ln  e
a b
ln e  ln e 
ab 
a
b
ln(e
a b
)
a b
a b
 ln x is one-to-one
Laws of Logarithms
Inverse Functions
Inverse Functions
5.4 Exponential Functions:
Differentiation and Integration
Properties of the Natural Exponential Function
1. The domain of f ( x)  ex is  -,   , and the range is  0,  .
2. The function f ( x)  e is continuous, increasing, and
one-to-one.
x
3. The graph of f ( x)  e x is concave upward on its entire domain.
4. lim e x  0 and lim e x  
x  
x 
5.4 Exponential Functions:
Differentiation and Integration
Let u be a differentiable function of x.
d u
d x
u du
x
e   e 
e   e 2.
1.
dx
dx
dx
ln e  x
Prove: #1
d
d
x
u '  dx ln e   dx  x
Chain
Rule
x
u
1 d x
 e   1
x
e dx
Identity
Differentiate both sides
d x
x
e   e

dx
5.4 Exponential Functions:
Differentiation and Integration
d 2 x 1
e   ?
dx
e
2 x 1
 2   2e
2 x 1
d 3 / x
 e   ?
dx
d  3 x1 
e


dx 
3
3 / x
e
 2 
x
3/ x
3e
2
x
Find the relative extrema of f ( x)  xe .
x
f '( x)  (1)e  xe
x
 e  x  1  x  1 is a critical #
x
x
 1, e  is a relative min. (1st derivative test)
1
1 x  
x  2
1  x  
x0
Sign of f '( x)
negative
positive
positive
Conclusion
Decreasing
Increasing
Increasing
Interval
  x  1
Test Value
x2
5.4 Exponential Functions:
Differentiation and Integration
 1, e 
1
5.4 Exponential Functions:
Differentiation and Integration
THM 5.12 Integration Rules for Exponential Functions
Let u be a differential function of x.
1.  e x dx  e x  C
2.  eu du  eu  C
2.  e du 
u
5.4 Exponential Functions:
Differentiation and Integration
e
3 x 1
dx  ?
u  3x  1 du  3dx
1 u
  e du
3
1 u
 e  C 
3
1 3 x 1
 e C
3
 5xe
 x2
dx  ?
2
u   x du  2 xdx
5 u

e du

2
5  x2

e C
2
5.4 Exponential Functions:
Differentiation and Integration
1/ x
e
dx

?
2
x
1
1
u
du   2 dx
x
x
   e du
u
 e
1/ x
C
 sin xe
cos x
dx  ?
u  cos x du   sin xdx
   e du
u
 e
cos x
C
5.4 Exponential Functions:
Differentiation and Integration

1
0
x
e dx  ?
u  x du  dx
1
0
  e du   e du
u
1
0
0
  e 
1
u
 e  e
0
1
 1 
e
1

0.632
u
5.4 Exponential Functions:
Differentiation and Integration

0
1
e cos  e  dx  ?
x
x
1
  1 cos u du  ?
e
 sin1  sin e
1

u  e du  e dx
x
 sin u e1
1
0.482
x
5.4 Exponential Functions:
Differentiation and Integration
Page. 356
33,34, 35 – 51 odd, 57, 58, 65, 69, 85 – 91odd,
99-105 odd