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Understanding Natural Exponential Functions: Differentiation and Integration

This lesson focuses on the properties and operations of natural exponential functions, specifically their differentiation and integration. The natural exponential function, denoted as f(x) = e^x, is defined as the inverse of the natural logarithm. Key concepts include solving exponential and logarithmic equations, understanding the function's domain and range, and identifying relative extrema and points of inflection. Students will also engage in classwork and homework exercises to reinforce their understanding of these pivotal calculus topics.

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Understanding Natural Exponential Functions: Differentiation and Integration

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  1. 5-4 Exponential Functions: Differentiation and IntegrationObjective: Differentiate and integrate natural exponential functions. Ms. Battaglia AP Calculus

  2. Definition of the Natural Exponential Function The inverse function of the natural logarithmic function f(x)=lnx is called the natural exponential function and is denoted by f -1(x) = ex That is, y = ex if and only if x = lny Inverse Relationship ln(ex) = x and elnx = x

  3. Solving Exponential Equations Solve 7 = ex+1

  4. Solving a Logarithmic Equation Solve ln(2x – 3) = 5

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

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

  7. Derivatives of the Natural Exponential Function Let u be a differentiable function of x.

  8. Differentiating Exponential Functions a. b.

  9. Locating Relative Extrema Find the relative extrema of f(x)=xex.

  10. The Standard Normal Probability Density Function Show that the standard normal probability density function has points of inflection when x=1, -1

  11. Shares Traded The numbers y of shares traded (in millions) on the NY Stock Exchange from 1990 through 2005 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 2000?

  12. Integration Rules for Exponential Functions Let u be a differentiable function of x. 1. 2.

  13. Integrating Exponential Functions Find

  14. Integrating Exponential Functions Find

  15. Integrating Exponential Functions a. b.

  16. Finding Areas Bounded by Exponential Functions a. b. c.

  17. Classwork/Homework • AB: Read 5.4 Page 358 #3, 4, 11-14, 37, 38, 39-57 odd • BC: HW: Read 5.4 Page 358 #1-11 odd, 39-59 odd,63,69,79,99, 103,105,111,121,125

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