Unformatted text preview: 5.1 The Natural Logarithm Introduction Study 5.1 p. 331 # 7, 9, 11, 15, 1927, 3143, 49, 51, 5763, 6973, 77, 80
Background: Integrate the following. Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 G. Battaly, 2011 1 5.1 The Natural Logarithm Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: Intro (2 of 29) 5.1 The Natural Logarithm Introduction Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: Intro (3 of 29) 5.1 The Natural Logarithm
What about ? Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: What about? (4 of 29) 5.1 The Natural Logarithm
What about ? Think: What function has a derivative = 1/x? Remember the 2nd FTC? Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: The Why (5 of 29) 4.4 FTC, Part 2: MVT, Ave. Value, 2nd FTC From Calc 1, Section 4.4 Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page Homework Part 1 Homework Part 2 Title: 2nd FTC (6 of 29) 5.1 The Natural Logarithm OR: So, we need to define a function that has a derivative = 1/x.
Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: Need a function (7 of 29) 5.1 The Natural Logarithm Definition of the natural logarithm What is its domain? What is its range? What does graph look like?
Need to investigate y = 1/x
Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: Definition of ln x (8 of 29) 5.1 The Natural Logarithm If x > 1, ln x ? 0 If 0 < x < 1, ln x ? 0
Because of Vertical Asymptote, limit domain to x > 0 Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: explore (9 of 29) 5.1 The Natural Logarithm Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: plot (10 of 29) 5.1 The Natural Logarithm Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: sketch ln x (11 of 29) 5.1 The Natural Logarithm Properties of y = ln x
1. Domain: x > 0 Range: all y 2. continuous, increasing over all domain 3. concave down Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: properties of ln x (12 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: example (13 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: example (14 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: example (15 of 29) (6x) approaches 0 from the right X is approaching 6 from the left Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: limits (16 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: derivative of ln x (17 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: example (18 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: example (19 of 29) Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: example (20 of 29) Point slope form of straight line Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: eq of tangent line (21 of 29) 5.1 The Natural Logarithm
The Derivative of the Natural Logarithm of the Absolute Value since: and: Note: u 0 Class Notes: Prof. G. Battaly, Westchester Community College, NY
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Homework Part 1 Homework Part 2 Title: ln(absolute value) (22 of 29) Title: Jan 286:01 PM (23 of 29) Title: Jan 286:08 PM (24 of 29) Title: Jan 286:15 PM (25 of 29) Title: Jan 286:18 PM (26 of 29) Title: Jan 286:24 PM (27 of 29) Title: Jan 286:34 PM (28 of 29) Title: Jan 286:42 PM (29 of 29) ...
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 Calculus, Natural logarithm, Logarithm, Westchester Community College, Calculus Home Page

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