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Miss Battaglia AP Calculus

5.1 The Natural Logarithmic Function: Differentiation (Day 2) Objective: Develop and use properties of natural log function and find derivatives, define e. Miss Battaglia AP Calculus. Def of Natural Logarithmic Function. The natural logarithmic function is defined by

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Miss Battaglia AP Calculus

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  1. 5.1 The Natural Logarithmic Function: Differentiation (Day 2)Objective: Develop and use properties of natural log function and find derivatives, define e. Miss Battaglia AP Calculus

  2. Def of Natural Logarithmic Function The natural logarithmic function is defined by The domain of the natural logarithmic function is the set of all positive real numbers.

  3. Properties of the Natural Log Function The natural log function has the following properties. • The domain is (0,∞) and the range is (-∞,∞) • The function is continuous, increasing, and one-to-one • The graph is concave down.

  4. Logarithmic Properties If a and b are positive numbers and n is rational, then the following properties are true. • ln(1) = 0 • ln(ab) = lna + lnb • ln(an) = nlna • ln(a/b) = lna – lnb

  5. Expanding Log Expressions a. b.

  6. Expanding Log Expressions c. d.

  7. The number e The letter e denotes the positive real number such that

  8. Evaluating Natural Log Expressions a. ln 45 b. ln 8.3 c. ln 0.6

  9. The Derivative of the Natural Log Function Let u be a differentiable function of x. 1. 2.

  10. Differentiation of Log Functions a. b.

  11. Differentiation of Log Functions c. d.

  12. Logarithmic Properties as Aids of Differentiation Differentiate

  13. Log Properties as Aids of Differentiation Differentiate

  14. Logarithmic Differentiation Find the derivative of

  15. Derivative Involving Absolute Value If u is a differentiable function of x such that u≠0, then

  16. Derivative Involving Absolute Value Find the derivative of

  17. Finding Relative Extrema Locate the relative extrema of

  18. Classwork/Homework AB: Read 5.1 Page 331 #11-17 odd, 47-61 odd, 83, 91, 101, 102

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