Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate...
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Transcript of Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate...
![Page 1: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/1.jpg)
Exponential Functions and Models
Lesson 3.1
![Page 2: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/2.jpg)
Contrast
LinearFunctions
Change at a constant rate Rate of change (slope) is a constant
ExponentialFunctions
Change at a changing rate Change at a constant percent rate
![Page 3: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/3.jpg)
Contrast
Suppose you have a choice of two different jobs at graduation Start at $30,000 with a 6% per year increase Start at $40,000 with $1200 per year raise
Which should you choose? One is linear growth One is exponential growth
![Page 4: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/4.jpg)
Which Job?
How do we get each nextvalue for Option A?
When is Option A better? When is Option B better?
Rate of increase a constant $1200
Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.06
Year Option A Option B
1 $30,000 $40,000
2 $31,800 $41,200
3 $33,708 $42,400
4 $35,730 $43,600
5 $37,874 $44,800
6 $40,147 $46,000
7 $42,556 $47,200
8 $45,109 $48,400
9 $47,815 $49,600
10 $50,684 $50,800
11 $53,725 $52,000
12 $56,949 $53,200
13 $60,366 $54,400
14 $63,988 $55,600
![Page 5: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/5.jpg)
Example
Consider a savings account with compounded yearly income You have $100 in the account You receive 5% annual interest
At end of year
Amount of interest earnedNew balance in
account
1 100 * 0.05 = $5.00 $105.00
2 105 * 0.05 = $5.25 $110.25
3 110.25 * 0.05 = $5.51 $115.76
4
5
View completed table
![Page 6: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/6.jpg)
Compounded Interest
Completed table
At end of year
Amount of interest earned
New balance in account
0 0 $100.001 $5.00 $105.002 $5.25 $110.253 $5.51 $115.764 $5.79 $121.555 $6.08 $127.636 $6.38 $134.017 $6.70 $140.718 $7.04 $147.759 $7.39 $155.1310 $7.76 $162.89
![Page 7: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/7.jpg)
Compounded Interest Table of results from
calculator Set Y= screen
y1(x)=100*1.05^x Choose Table (♦ Y)
Graph of results
![Page 8: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/8.jpg)
Compound Interest
Consider an amount A0 of money deposited in an account Pays annual rate of interest r percent Compounded m times per year Stays in the account n years
Then the resulting balance An
0 1m n
n
rA A
m
![Page 9: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/9.jpg)
Exponential Modeling
Population growth often modeled by exponential function
Half life of radioactive materials modeled by exponential function
![Page 10: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/10.jpg)
Growth Factor
Recall formulanew balance = old balance + 0.05 * old balance
Another way of writing the formulanew balance = 1.05 * old balance
Why equivalent?
Growth factor: 1 + interest rate as a fraction
![Page 11: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/11.jpg)
Decreasing Exponentials Consider a medication
Patient takes 100 mg Once it is taken, body filters medication out over
period of time Suppose it removes 15% of what is present in
the blood stream every hour
At end of hour Amount remaining
1 100 – 0.15 * 100 = 85
2 85 – 0.15 * 85 = 72.25
3
4
5
Fill in the rest of the
table
What is the growth factor?
![Page 12: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/12.jpg)
Decreasing Exponentials Completed chart
Graph
At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06
Amount Remaining
0.00
20.00
40.00
60.00
80.00
100.00
0 1 2 3 4 5 6 7 8
At End of Hour
Mg
rem
ain
ing
Growth Factor = 0.85
Note: when growth factor < 1, exponential is a decreasing
function
![Page 13: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/13.jpg)
Solving Exponential Equations Graphically
For our medication example when does the amount of medication amount to less than 5 mg
Graph the functionfor 0 < t < 25
Use the graph todetermine when
( ) 100 0.85 5.0tM t
![Page 14: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/14.jpg)
General Formula
All exponential functions have the general format:
Where A = initial value B = growth rate t = number of time periods
( ) tf t A B
![Page 15: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/15.jpg)
Typical Exponential Graphs
When B > 1
When B < 1
( ) tf t A B
![Page 16: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/16.jpg)
Using e As the Base
We have used y = A * Bt
Consider letting B = ek
Then by substitution y = A * (ek)t
Recall B = (1 + r) (the growth factor)
It turns out that k r
![Page 17: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/17.jpg)
Continuous Growth
The constant k is called the continuous percent growth rate
For Q = a bt k can be found by solving ek = b
Then Q = a ek*t
For positive a if k > 0 then Q is an increasing function if k < 0 then Q is a decreasing function
![Page 18: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/18.jpg)
Continuous Growth
For Q = a ek*t Assume a > 0
k > 0
k < 0
![Page 19: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/19.jpg)
Continuous Growth
For the functionwhat is thecontinuous growth rate?
The growth rate is the coefficient of t Growth rate = 0.4 or 40%
Graph the function (predict what it looks like)
0.43 tQ e
![Page 20: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/20.jpg)
Converting Between Forms
Change to the form Q = A*Bt
We know B = ek
Change to the form Q = A*ek*t
We know k = ln B (Why?)
0.43 tQ e
94.5(1.076)tQ
![Page 21: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/21.jpg)
Continuous Growth Rates
May be a better mathematical model for some situations
Bacteria growth Decrease of medicine
in the bloodstream
Population growth of a large group
![Page 22: Exponential Functions and Models Lesson 3.1. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions.](https://reader035.fdocuments.in/reader035/viewer/2022081603/56649e105503460f94afb1bd/html5/thumbnails/22.jpg)
Example A population grows from its initial level of
22,000 people and grows at a continuous growth rate of 7.1% per year.
What is the formula P(t), the population in year t? P(t) = 22000*e.071t
By what percent does the population increase each year (What is the yearly growth rate)? Use b = ek