Miss Battaglia AP Calculus. The natural logarithmic function is defined by The domain of the natural...
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Transcript of Miss Battaglia AP Calculus. The natural logarithmic function is defined by The domain of the natural...
5.1 The Natural Logarithmic Function: Differentiation
Objective: Develop and use properties of natural log function and find derivatives, define e.
Miss BattagliaAP Calculus
The natural logarithmic function is defined by
The domain of the natural logarithmic function is the set of all positive real numbers.
Def of Natural Logarithmic Function
The natural log function has the following properties.
1. The domain is (0,∞) and the range is (-∞, ∞)
2. The function is continuous, increasing, and one-to-one
3. The graph is concave down.
Properties of the Natural Log Function
If a and b are positive numbers and n is rational, then the following properties are true.
1. ln(1) = 02. ln(ab) = ln a + ln b3. ln(an) = n ln a4. ln(a/b) = ln a – ln b
Logarithmic Properties
a. b.
Expanding Log Expressions
c. d.
Expanding Log Expressions
The letter e denotes the positive real number such that
The number e
a. ln 2
b. ln 32
c. ln 0.1
Evaluating Natural Log Expressions
Let u be a differentiable function of x.
1. 2.
The Derivative of the Natural Log Function
a. b.
Differentiation of Log Functions
c. d.
Differentiation of Log Functions
Differentiate
Logarithmic Properties as Aids of Differentiation
Differentiate
Log Properties as Aids of Differentiation
Find the derivative of
Logarithmic Differentiation
If u is a differentiable function of x such that u≠0, then
Derivative Involving Absolute Value
Find the derivative of
Derivative Involving Absolute Value
Locate the relative extrema of
Finding Relative Extrema
BC: Read 5.1 Page 331 #11-17 odd, 47-61 odd, 83, 91, 101, 102
Classwork/Homework