# EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

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EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION
Section 5.4

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When you are done with your homework, you will be able to…
Develop properties of the natural exponential function Differentiate natural exponential functions Integrate natural exponential functions

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Definition of the Natural Exponential Function
The inverse function of the natural logarithmic function is called the natural exponential function and is denoted by That is,

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The inverse relationship between the natural logarithmic function and the natural exponential function can be summarized as follows:

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Solve 6.0 0.0

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Solve All of the above. B and C

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Solve. Round to the nearest ten thousandth.
0.680 0.0001

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Theorem: Operations with Exponential Functions
Let a and b be any real numbers.

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Properties of the Natural Exponential Function
The domain is all real numbers and the range is all positive real numbers The natural exponential function is continuous, increasing, and one-to-one on its entire domain. The graph of the natural exponential function is concave upward on its entire domain. The limit as x approaches negative infinity is 0 and the limit as x approaches positive infinity is infinity.

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Theorem: Derivative of the Natural Exponential Function
Let u be a differentiable function of x.

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Find the derivative of

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Find the derivative of

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Find the derivative of

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Theorem: Integration Rules for Exponential Functions

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Evaluate

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Evaluate

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