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Exponential Functions
©G. Battaly, 2013 1
May 31, 2013
Intro
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
Assignment:5.4 # 115, 2528 all, 29, 37, 39; 41, 45, 49, 51, 53, 57, 6369, 73, 79; 99, 103, 107, 111, 115, 117, 121
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Exponential Functions
©G. Battaly, 2013 2
May 31, 2013
exponential functions
5.4 Exponential Functions: Differentiation and Integration
Bacteria reproduce by binary fission. A single cell divides to form 2 cells.
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
# 2x
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Exponential Functions
©G. Battaly, 2013 3
May 31, 2013
exponential functions
5.4 Exponential Functions: Differentiation and Integration
# 2xBacteria reproduce by binary fission.
Consider a single cell:
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
y = 2x is exponential function.
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Exponential Functions
©G. Battaly, 2013 4
May 31, 2013
Feb 411:35 AM
Exponential Function
This is base 2.Have you seen functions with a different base?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
x # 2x
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Exponential Functions
©G. Battaly, 2013 5
May 31, 2013
Intro
5.4 Exponential Functions: Differentiation and Integration
Natural Exponential Function
y = ln x
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 6
May 31, 2013
properties of e^x
y = e x
y = e x
Properties of the Natural Exponential Function
Domain?
Range?
Continuous?
1to1?
Concave up or down?
y = e x
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 7
May 31, 2013
properties of e^x
y = e x
y = e x
Properties of the Natural Exponential Function
Domain?
Range?
Continuous?
1to1?
Concave up or down?
y = e x
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 8
May 31, 2013
properties of e^x
y = e x
y = e x
Properties of the Natural Exponential Function
Domain?
Range?
Continuous?
1to1?
Concave up or down?
y = e x
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 9: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/9.jpg)
Exponential Functions
©G. Battaly, 2013 9
May 31, 2013
solving exponential equations
5.4 Exponential Functions: Differentiation and Integration
Solving equations with variable exponents
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 10
May 31, 2013
solving exponential equations
5.4 Exponential Functions: Differentiation and Integration
Solving equations with variable exponents
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
2 approaches: 1. find ln of both sides 2. convert to logarithmic form
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Exponential Functions
©G. Battaly, 2013 11
May 31, 2013
solving exponential equations
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
F: solve for x
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Exponential Functions
©G. Battaly, 2013 12
May 31, 2013
solving exponential equations
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
2 approaches: 1. find ln of both sides 2. convert to logarithmic form
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Exponential Functions
©G. Battaly, 2013 13
May 31, 2013
solving logarithmic equations
5.4 Exponential Functions: Differentiation and Integration
The log is the exponent.
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 14
May 31, 2013
solving logarithmic equations
5.4 Exponential Functions: Differentiation and Integration
The log is the exponent.
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 15: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/15.jpg)
Exponential Functions
©G. Battaly, 2013 15
May 31, 2013
Feb 47:15 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 16: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/16.jpg)
Exponential Functions
©G. Battaly, 2013 16
May 31, 2013
Feb 47:15 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 17: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/17.jpg)
Exponential Functions
©G. Battaly, 2013 17
May 31, 2013
solving logarithmic equations
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 18: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/18.jpg)
Exponential Functions
©G. Battaly, 2013 18
May 31, 2013
solving logarithmic equations
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
compare answers: How are they different? Which is correct for the original problem? (Sketch y1 = ln x2 and y2 = 10 to find out.)
![Page 19: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/19.jpg)
Exponential Functions
©G. Battaly, 2013 19
May 31, 2013
derivative of e^x
5.4 Exponential Functions: Differentiation and Integration
Finding the derivative......
?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 20
May 31, 2013
derivative of e^u
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 21: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/21.jpg)
Exponential Functions
©G. Battaly, 2013 21
May 31, 2013
derivative of e^u
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 22: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/22.jpg)
Exponential Functions
©G. Battaly, 2013 22
May 31, 2013
example
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 23
May 31, 2013
example
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 24: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/24.jpg)
Exponential Functions
©G. Battaly, 2013 24
May 31, 2013
example
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
![Page 25: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/25.jpg)
Exponential Functions
©G. Battaly, 2013 25
May 31, 2013
example
5.4 Exponential Functions: Differentiation and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
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Exponential Functions
©G. Battaly, 2013 26
May 31, 2013
Feb 47:35 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 27: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/27.jpg)
Exponential Functions
©G. Battaly, 2013 27
May 31, 2013
Feb 47:35 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 28: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/28.jpg)
Exponential Functions
©G. Battaly, 2013 28
May 31, 2013
Feb 47:41 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 29: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/29.jpg)
Exponential Functions
©G. Battaly, 2013 29
May 31, 2013
Feb 47:41 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 30: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/30.jpg)
Exponential Functions
©G. Battaly, 2013 30
May 31, 2013
Feb 611:40 AM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
Example of Exponential Function used in Statistics
Standard Normal Probability Density Function has mean of 0 and Inflection point at + 1σ
Find the inflection points if σ=1 (See textbook to check your solution)
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Exponential Functions
©G. Battaly, 2013 31
May 31, 2013
Feb 611:40 AM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 32: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/32.jpg)
Exponential Functions
©G. Battaly, 2013 32
May 31, 2013
Feb 611:40 AM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 33: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/33.jpg)
Exponential Functions
©G. Battaly, 2013 33
May 31, 2013
Feb 611:40 AM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 34: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/34.jpg)
Exponential Functions
©G. Battaly, 2013 34
May 31, 2013
May 308:35 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 35: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/35.jpg)
Exponential Functions
©G. Battaly, 2013 35
May 31, 2013
May 308:35 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 36: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/36.jpg)
Exponential Functions
©G. Battaly, 2013 36
May 31, 2013
May 308:37 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration
![Page 37: 41, 45, 49, 51, 53, 57, 6369, 115, 117, 121 - Battaly...solving logarithmic equations 5.4 Exponential Functions: Differentiation and Integration Calculus Home Page Class Notes: Prof](https://reader035.vdocuments.site/reader035/viewer/2022081601/60f7dfaff1cad15e12607480/html5/thumbnails/37.jpg)
Exponential Functions
©G. Battaly, 2013 37
May 31, 2013
May 308:37 PM
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1 Homework Part 2
5.4 Exponential Functions: Differentiation and Integration