5.1 the natural logarithm - battaly · 5.1 the natural logarithm introduction calculus home page...
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Natural Logarithm: Definition and Derivative
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May 28, 2013
Background
5.1 The Natural Logarithm Introduction
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Background: Integrate the following.
Study 5.1; p. 331 # 7, 9, 11, 15, 1927, 3143, 49, 51, 5763, 6973, 77, 80, 83
Natural Logarithm: Definition and Derivative
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Background
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Background: Integrate the following.
Natural Logarithm: Definition and Derivative
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Intro
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Background Summary
Natural Logarithm: Definition and Derivative
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What about?
5.1 The Natural Logarithm
What about ?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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Think Antiderivative
5.1 The Natural Logarithm
What about ?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Remember the 2nd FTC?
Think: What function has a derivative = 1/x?
Natural Logarithm: Definition and Derivative
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2nd FTC
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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4.4 FTC, Part 2: MVT, Ave. Value, 2nd FTC From Calc 1, Section 4.4
2nd FTCIf f is continuous on an open interval I containing a,
then, for every x on the interval:
What about ?
Natural Logarithm: Definition and Derivative
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Need a function
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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OR:
So, to find we need to define afunction that has a derivative = 1/x.
What about ?
Natural Logarithm: Definition and Derivative
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Definition of ln x
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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Definition of the natural logarithm
What is its domain?What is its range?
What does graph look like?
Need to investigate y = 1/x
Natural Logarithm: Definition and Derivative
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explore
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
If x > 1, ln x ? 0If 0 < x < 1, ln x ? 0
Because of Vertical Asymptote,limit domain to x > 0
x
web; change f to 1/xthen change b
Natural Logarithm: Definition and Derivative
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May 28, 2013
plot
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
0.5
Investigation:Let x = 0.5, 1, 1.5, ....Use calculator to find the integral of y=1/x from 1 to 0.5, from 1 to 1.5, etc.Plot points (0.5, ln 0.5), (1, ln 1), (1.5, ln 1.5), etc.
Natural Logarithm: Definition and Derivative
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sketch ln x
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
0.5
Result is y = ln x
Natural Logarithm: Definition and Derivative
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May 28, 2013
properties of ln x
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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Properties of y = ln x
1. Domain: x > 0 Range: all y
2. continuous, increasing over all domain
3. concave down
Natural Logarithm: Definition and Derivative
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example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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Remember Properties of Logarithms ...
1. ln (1) = 0 2. ln (e) = 1
3. ln (a b) = ln(a) + ln(b)
4. ln a = ln(a) ln(b) b
5. ln an = n ln(a)
Natural Logarithm: Definition and Derivative
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example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Remember Properties of Logarithms ...
Natural Logarithm: Definition and Derivative
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example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Remember Properties of Logarithms ...
Natural Logarithm: Definition and Derivative
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example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Write as the natural log of a single expression.
Natural Logarithm: Definition and Derivative
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example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Write as the natural log of a single expression.
Natural Logarithm: Definition and Derivative
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limits
X is approaching 6 from the left
(6x) approaches 0 from the right
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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May 28, 2013
limits
X is approaching 6 from the left
(6x) approaches 0 from the right
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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May 28, 2013
derivative of ln x
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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Derivative of the Natural Logarithm: Use the 2nd FTC
d (ln x) = dx 1
By definition:
d (ln x) = 1 dx x
d (ln u) = 1 du = u' dx u dx u
Natural Logarithm: Definition and Derivative
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example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
OR:
Natural Logarithm: Definition and Derivative
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May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
©Battaly, 2013 27
May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
©Battaly, 2013 28
May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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eq of tangent line
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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Natural Logarithm: Definition and Derivative
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eq of tangent line
Point slope form of straight line
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
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ln(absolute value)
5.1 The Natural Logarithm
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
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The Derivative of the Natural Logarithm of the Absolute Value
since:
and:
Note: u ≠0
Natural Logarithm: Definition and Derivative
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May 28, 2013
Jan 286:01 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
HW problems
Natural Logarithm: Definition and Derivative
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May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
©Battaly, 2013 34
May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Natural Logarithm: Definition and Derivative
©Battaly, 2013 35
May 28, 2013
example
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2