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**Unformatted text preview: **Exponential
Functions:
Differentiation &
Integration
Integration
Section 5.4 Defn. of the Natural
Exponential Function
Exponential The inverse of the natural
logarithmic function f(x) = ln x is
called the natural exponential
function : f –1 (x) = ex
y = ex iff x = ln y f –1(x) = ex f(x) = ln x ln(ex)
eln x
ln =x =x e=1 Operations with Exponential
Functions
Functions
1. e e =e
ab a a +b e
a −b
=e
b
e Properties of the Natural
Exponential Function
Exponential
1. The domain of f(x) = ex is ( − ∞ , ∞).
2. The function f(x) = ex is continuous,
increasing, and one-to-one on its
entire domain.
x
3. The graph of f(x) = e is concave
upward on its entire domain. lim e = 0 and lim e = ∞
x x → −∞ x x →∞ The Derivative of the Natural
Exponential Function
Exponential
Let u be a differentiable function of x.
1. d e x = e x
dx 2. du
u du
e =e
dx
dx Integration Rules for
Exponential Functions
Exponential
Let u be a differentiable function of x.
x
x
1.
e dx = e + c
2. ∫
e du = e
∫
u u +c Joke Time
Joke How can you stop a skunk from
smelling?
Hold its nose Why do golfers wear 2 pairs of
pants?
In case they get a hole in one ...

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