Math 162
Class 4
Natural Logarithm
Natural Logarithm
Definition:
Domain:
If
The Natural Exponential Function
It has an inverse function.
We will call it the natural exponential function, denoted as
Domain of
Range of
What is the derivative of
Derivative of Inverse Functions
Derivative of an Inverse Function
Theorem: If a function
then
Example
Let
Example
Let
The Exponential Function
Let
Derivative of Exponential Function
The exponential function is differentiable everywhere and
Since
Since
The Natural Base
Define
Suppose
Applying the exponential function on both sides:
Since
This gives us
If
For a rational number
Irrational Powers
If
Then
General powers:
Suppose
Other Properties of
Let
Limits
Let
As
Therefore, as
As
Therefore, as
Summary
Domain of
Range of
Inverse Sine
Inverse Tangent
Derivatives
Antiderivatives
Math 162 Class 4