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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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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 The domain of the natural logarithmic function is the set of all positive real numbers.
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.
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
The number e The letter e denotes the positive real number such that
Evaluating Natural Log Expressions a. ln 45 b. ln 8.3 c. ln 0.6
The Derivative of the Natural Log Function Let u be a differentiable function of x. 1. 2.
Logarithmic Properties as Aids of Differentiation Differentiate
Log Properties as Aids of Differentiation Differentiate
Logarithmic Differentiation Find the derivative of
Derivative Involving Absolute Value If u is a differentiable function of x such that u≠0, then
Derivative Involving Absolute Value Find the derivative of
Finding Relative Extrema Locate the relative extrema of
Classwork/Homework AB: Read 5.1 Page 331 #11-17 odd, 47-61 odd, 83, 91, 101, 102