miss battaglia ap calculus
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5.1 The Natural Logarithmic Function: Differentiation 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 - PowerPoint PPT PresentationTRANSCRIPT
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5.1 The Natural Logarithmic Function: Differentiation
Objective: Develop and use properties of natural log function and find derivatives, define e.
Miss BattagliaAP Calculus
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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
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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
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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
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a. b.
Expanding Log Expressions
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c. d.
Expanding Log Expressions
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The letter e denotes the positive real number such that
The number e
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a. ln 2
b. ln 32
c. ln 0.1
Evaluating Natural Log Expressions
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Let u be a differentiable function of x.
1. 2.
The Derivative of the Natural Log Function
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a. b.
Differentiation of Log Functions
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c. d.
Differentiation of Log Functions
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Differentiate
Logarithmic Properties as Aids of Differentiation
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Differentiate
Log Properties as Aids of Differentiation
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Find the derivative of
Logarithmic Differentiation
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If u is a differentiable function of x such that u≠0, then
Derivative Involving Absolute Value
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Find the derivative of
Derivative Involving Absolute Value
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Locate the relative extrema of
Finding Relative Extrema
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BC: Read 5.1 Page 331 #11-17 odd, 47-61 odd, 83, 91, 101, 102
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