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Transcript of 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural...
![Page 1: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/1.jpg)
5-1: Natural Logs and Differentiation
Objectives:
©2003Roy L. Gover (www.mrgover.com)
•Review properties of natural logarithms•Differentiate natural logarithm functions
![Page 2: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/2.jpg)
Review
ln yx y e x
n logl ex x
2.718281828e
![Page 3: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/3.jpg)
ExampleWrite as a natural logarithm:
2 7.3891e 1
2 1 87 .64e
![Page 4: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/4.jpg)
ExampleWrite in exponential form:
ln 7.3891 2
1ln1.6487
2
![Page 5: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/5.jpg)
Properties of Logarithms
ln(1)=0
ln(ab)=ln a + ln b
ln an=n ln a
ln ln lna
a bb
ln e=1
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•What is the domain?
•What is the range?
Important Idea
•Wassup at x=1?
The graph of the natural log function looks like:
![Page 7: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/7.jpg)
Try ThisExpand the log function:
ln[( ( )] )x y x y
ln( ) ln( )x y x y
![Page 8: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/8.jpg)
Try ThisExpand the log function:
10ln
3
x
y
ln(10 ) ln(3 )x y then:
ln10 ln ln 3 lnx y
![Page 9: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/9.jpg)
Try ThisExpand the log function:
ln 2x
1ln(2 )
2x then:
1(ln 2 ln )
2x
![Page 10: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/10.jpg)
Try ThisEvaluate:
4ln e
4ln 4(1) 4e
![Page 11: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/11.jpg)
Try ThisEvaluate:
2 2ln ( )e x y
2 2x y
![Page 12: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/12.jpg)
Try ThisEvaluate:
ln 5
1.6094
![Page 13: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/13.jpg)
Try ThisWrite as a logarithm of a single quantity:
1ln 2ln
2x y
2ln x y
![Page 14: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/14.jpg)
Try ThisWrite as a logarithm of a single quantity:
n(3 1) ln 2l (2 )x x
3 1ln
2 2
x
x
![Page 15: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/15.jpg)
Try ThisFind the antiderivative:1
dxx
Can’t solve using the power rule
![Page 16: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/16.jpg)
Important Idea
•There exists an area
under the curve
equal to 1
1( )f x
x
ln 1e •
• 1b
adx
x is an area under
1( )f x
x
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Definition
1 2.72e
1( )f x
xe is the
positive real number such that:
1
1ln 1
ee dt
t
Area = 1
![Page 18: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/18.jpg)
Definitionfrom the previous definition...
1 1[ln ]
x
a
d dx dt
dx dx t x
therefore:
1[ln ]
dx
dx x
memorize
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The chain rule version:
1[ln ]
d duu
dx u dx
Definition
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Examples
[ln 4 ] d
xdx
2[ln(3 1)]d
xdx
![Page 21: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/21.jpg)
Try This
3[ln( 1 ] 4 )d
xdx
2
3
12
4 1
x
x
![Page 22: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/22.jpg)
Example
[ ln ]d
x xdx
Hint: use the product rule
![Page 23: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/23.jpg)
Example
lnd x
dx x
Hint: use the quotient rule
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Example
ln1
d x
dx x
Is this a quotient rule problem?
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Example
2[ln ]d
xdx
2[(ln ) ]d
xdx
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Try This
2 3( ) 2(ln )f x x x
Find the derivative:
Hint: Rewrite using log properties then use chain rule
![Page 27: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/27.jpg)
Solution
3
3 3
3
2(2 ln )
2(2 )(ln )
= 16(ln )
x x
x x
x x
Rewrite:2 3( ) 2(ln )f x x x
![Page 28: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/28.jpg)
SolutionUse chain rule:
2
2
1'( ) 1 16(3)(ln )
48(ln ) =
1
f x xx
x
x
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Try This
3
2 2
2 1( ) ln
( 1)
xf x
x x
Rewrite using log properties before differentiation...
![Page 30: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/30.jpg)
3 2 2
3 2 2
3 2
( ) ln 2 1 ln[ ( 1) ]
1 ln(2 1) [ln ln( 1) ]
21
ln(2 1) ln 2ln( 1)2
f x x x x
x x x
x x x
Rewrite:Solution
![Page 31: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/31.jpg)
23 2
1 1 1 1 '( ) (6
) 2 (2 )2 2 1 1
f x x xxx x
…then differentiate
Solution
2
3 2
3 1 4
2 1 1
x x
xx x
And simplify:
![Page 32: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/32.jpg)
DefinitionSince ln x is not defined for negative values of x, you may frequently see ln|x|. The absolute value rule for ln is: 1 '
ln ln 'd d u
u u udx dx u u
When differentiating a logarithm, you may ignore any absolute value sign.
![Page 33: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/33.jpg)
Try This
( ) ln sin(2 )f x xFind the derivative:
1cos(2 )(2) 2cot(2 )
sin(2 )x x
x
Don’t forget the chain rule
![Page 34: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/34.jpg)
Try ThisFind the equation of the line tangent to:
lny x x at (1,1)
2 1y x
![Page 35: 5-1: Natural Logs and Differentiation Objectives: ©2003Roy L. Gover () Review properties of natural logarithms Differentiate natural logarithm.](https://reader034.fdocuments.us/reader034/viewer/2022051316/5697bfa81a28abf838c994ab/html5/thumbnails/35.jpg)
Lesson CloseThe natural logarithm is frequently used in Calculus. Be certain that you understand the properties of logarithms and know how to differentiate and integrate (next section) logarithmic functions.
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Assignment
1. 324/15-29 Odd (Slides 1-14)
2. 324/31-63 Odd (Slides 15-36)