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Ms. Battaglia AB Calculus

5-4 Exponential Functions: Differentiation and Integration (Day 2) Objective: Differentiate and integrate natural exponential functions. Ms. Battaglia AB Calculus. Definition of the Natural Exponential Function.

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Ms. Battaglia AB Calculus

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  1. 5-4 Exponential Functions: Differentiation and Integration (Day 2)Objective: Differentiate and integrate natural exponential functions. Ms. Battaglia AB 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 9 – 2ex = 7

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

  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 extremaand the points of inflection (if any exist) of the function. Use a calculator to confirm your results.

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

  11. Integrating Exponential Functions Find

  12. Integrating Exponential Functions Find

  13. Integrating Exponential Functions a. b.

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

  15. Classwork/Homework • AB: Page 360 #99-125 odd (skip 113, 115)

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