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AP Calculus AB Chapter 5, Section 4

AP Calculus AB Chapter 5, Section 4. Exponential Functions: Differentiation and Integration 2013 - 2014. The Natural Exponential Function. If x is a rational number, then the inverse of is . Definition of the Natural Exponential Function:

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AP Calculus AB Chapter 5, Section 4

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  1. AP Calculus ABChapter 5, Section 4 Exponential Functions: Differentiation and Integration 2013 - 2014

  2. The Natural Exponential Function • If x is a rational number, then the inverse of is . • 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, • if and only if

  3. The Inverse Relationship • Basically this happens:

  4. Solving Exponential Equations • Solve

  5. Solving a Logarithmic Equation • Solve

  6. Operations with Exponential Functions • Let a and b be any real number

  7. Properties of the Natural Exponential Function • The domain of is (-∞, ∞), and the range is (0, ∞). • The function is continuous, increasing, and one-to-one on its entire domain. • The graph of is concave upward on its entire domain. • and

  8. Derivatives of Exponential Functions • The natural exponential function is its own derivative. • Let u be a differentiable function of x.

  9. Differentiating Exponential Functions

  10. Locating Relative Extrema • Find the relative extrema of

  11. The Standard Normal Probability Density Function • Show that the standard normal probability density function • Has points of inflection when

  12. Shares Traded • The number y of shares traded (in millions) on the New York Stock Exchange from 1990 through 2002 can be modeled by Where t represents the year, with t = 0 corresponding to 1990. At what rate was the number of shares traded changing in 1998? (Source: New York Stock Exchange, Inc.)

  13. Integrals of Exponential Functions • Let u be a differentiable function of x.

  14. Integrating Exponential Functions • Find

  15. Integration Exponential Functions • Find

  16. Integrating Exponential Function

  17. Integrating Exponential Functions

  18. Finding Areas Bounded By Exponential Functions • Evaluate each definite integral:

  19. Ch. 5.4 Homework • Pg. 356 – 358, #’s: 7, 21 – 24, 33 (a & b), 35, 37, 43, 51, 57, 59, 61, 65, 69, 73, 77, 85 – 105 (every other odd)

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