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EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION. Section 5.4. 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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EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

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  1. EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION Section 5.4

  2. 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

  3. 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,

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

  5. Solve • 6.0 • 0.0

  6. Solve • All of the above. • B and C

  7. Solve. Round to the nearest ten thousandth. • 0.680 • 0.0001

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

  9. 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.

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

  11. Find the derivative of

  12. Find the derivative of

  13. Find the derivative of

  14. Theorem: Integration Rules for Exponential Functions

  15. Evaluate

  16. Evaluate

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