Download 5.4: Exponential Functions: Differentiation and Integration

AP CALCULUS-AB
Section Number:
LECTURE NOTES
Topics: Exponential Functions: Differentiation
MR. RECORD
Day: 1 of 2
5.4
From Section 5.1, we discussed the graph of y  ln x . In this section we focus on the inverse of f (x)  ln x
which is f 1 (x)  e x . Note their graphs below.
y
y  ex
2
y  ln x
1
Definition of Natural Exponential Function
The inverse function of the natural logarithm function
f (x)  ln x is called the natural exponential function
and is denoted by
f 1 (x)  e x
x
-3
-2
-1
1
2
3
-1
That is to say,
y  e x if and only if x  ln y
-2
-3
The inverse relationship between the natural logarithm function and the natural exponential function can be
summarized as follows:
ln(e x )  x and
eln x  x
Algebra Review
Example 1:
Solve each of the following equations.
a. Solve 7  e x 1
Theorem: Operations with Exponential Functions
Let a and b be any real numbers
ea
1. ea e b  ea  b
2. b  eab
e
b. ln(2 x  3)  5
Properties of the Natural Exponential Function
The natural exponential function has the following properties.
1. The domain is ( , ) and the range is (0, ) .
2. The function is continuous, increasing, and one-to-one.
3. The graph is concave upward.
4.
lim e x  0 and lim e x  
x 
x 
Derivatives of the Natural Exponential Function
The Derivative of the Natural Logarithmic Function
Let u be a differentiable function of x.
d x
d u
du
e   e x
e   eu
1.
2.
dx
dx
dx
Proof:
Example 2: Differentiate each of the following.
a.
d 2 x 1
e 
dx 
Example 3: a.
b.
d 3x
e 

dx 
Find the relative extrema of f (x)  xe x .
y
2
1
x
-3
b. Sketch the graph of f(x) to the right.
-2
-1
1
-1
-2
-3
2
3
AP CALCULUS-AB
Section Number:
LECTURE NOTES
Topics: Exponential Functions: Integration
MR. RECORD
Day: 2 of 2
5.4
Integrals of the Natural Exponential Function
The Integral of the Natural Logarithmic Function
Let u be a differentiable function of x.
e
1.
x
dx  e x  C
2.
e
u
du
dx  eu  C
dx
Example 4: Integrate each of the following.
a.
3 x 1
 e dx
x
 5xe dx
2
b.
1
ex
c.  2 dx
x
 sin x e
d.
cos x
dx
Example 5: Evaluate each of the following definite integrals.
1
a.
0
1
x
 e dx
ex
0 1  e x dx
b.
0
c.
 e
x
1
cos(e x )dx
y
y
y
1
1
1
x
1
x
1
x
-1