Math Modeling

1
Math Modeling

• Section 2.1

2
Introduction to Exponential Functions and Models

• Suppose that in an effort to improve classroom
  attendance, I offer to pay you each day for
  attending class! Being even more generous, I
  offer you a choice
• Choice 1 Receive 2 cents on the first day you
  attend class, 4 cents the second day, 8 cents the
  third day, and so on.
• Choice 2 Receive 1,000 per class attended.
• Before we go on, which choice do you choose?
  _____

3
Create a formula for the choice above that
represents linear growth, defining your input and
output variables.

• C2 1000X
• X the number of classes attended
• C2 The amount of money received from attending
  X classes

4
Draw a graph of the money you make over 30
classes. Label each axis.
C2(X) from attending class
30,000 20,000 10,000
(30,30000)
(20,20000)
(10,10000)
(0,0)
X classes
0 5 10 15 20 25 30
5
How much money would have after 30 classes?

• 30,000

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Exponential growth
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22
23
16 cents
24
32 cents
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C1 2x x number of days C1 money
received On the xth day
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Graph the exponential model for up to 30 classes.
Label each axis.
C1(X) Money from attending class (in cents)
(30,1073741824)
1,000,000,000 500,000,000
(20,1048576)
(0,0)
(10,1024)
X classes
0 5 10 15 20 25 30
8
How much money would have after 30 classes?

• C2(30) 30,000
• C1(30) 1073741824 cents
• Or 10, 737,418.24

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• As you learned in section 1.4, linear models
  increase at a constant rate of change. In this
  section we learn that exponential models increase
  at a constant percentage change. What is the
  constant percentage change of the exponential
  model above?
• How does the output values change from one class
  to another?

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The Exponential model
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• Is the following function an
• increasing exponential model
• or a decreasing exponential model?

Decreasing
2.Is the following function an increasing
exponential model or a decreasing exponential
model?
Neither
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Developing the concept of the exponential
Model Suppose we have two accounts Account 1
begins with 1000 and gets 100 added to it
every month. Account 2 begins with 1000 and
gets 10 added to it every month. Lets analyze
the growth of the two accounts. Account 1
1100
1200
1400
1300
Report a complete model for account 1
A(x) 100x 1000 x the number of months
since the start of the account A(x) Account
balance
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Account 2
Report a complete model for account 2
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Calculating Percentage Change

• To find the constant percentage change from an
  exponential formula, you use the following (b
  1)100

Example 1 From the example above . a.
Calculate (b 1)100 __________________________
(1.10 - 1)100 10
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b. Interpret the percentage change. (Similar to
the interpretation of the slope)

• Every month, the account balance increases by
  10.

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Example 2

• The total number of hours spent on line by U.S.
  residents in a
• certain state during 2004 can be described by the
  function
• H(m) 18(1.23m) million hours by the end of the
  mth month after
• the beginning of January 2004.
• Was the total number of hours spent on line by
  U.S. residents
• in this state growing or decaying during 2004?
• Find the constant percentage change for this
  situation.
• d) Interpret the constant percentage change in
  this context.

Growing
(b 1)100 (1.23 1)100 23
The number of hours spent online by U.S.
residents in a certain state is increasing by 23
each month.
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Example 3

• A toy that was expected to be a passing fad had
  sales of 40
• million in its first year. Its sales then
  declined by 32 each year.
• Find a model for the sales of the toy.
• b) Estimate the toys sales 3 years later.
• c) Find the percentage change of the model.

S(x) 40(.68)x x the number of years since
toy was introduced S(x) sales of the toy in
millions
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Perfectly Exponential data

• The following data represents the number of DVDs
  sold, in thousands, of a popular movie x months
  after its release.

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Show that the data is not linear by filling in
the following table and looking at a scatter
plot.
60 100 -40
36 60 -24
21.6 36 -14.4
12.9621.6 -8.64
7.77612.96 -5.184
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To determine if data (that is evenly spaced) is
exponential, we look at the percentage
differences as follows2nd output value in
interval 1st output value in interval
1st output value in the interval(and
then multiply times 100)

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60 100 100 -40
36 60 60 -40
21.6 - 36 36 -40
7.77612.96 12.96 -40
12.9621.6 21.6 -40
If the percentage differences are nearly
constant, then we can model the data with an
exponential function. T(x) is decreasing so this
is an example of ________________________.
Exponential Decay
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• Interpret the above percentage differences
• Write a model for the data.

Every month, the number of dvds sold is
decreasing by 40
T(x) 100(.60)x x the number of months after
the release of a popular movie T(x) the
number of DVDs sold (in thousands)
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Exponential Regression
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Can we model the data above by an exponential
function?
38 21 21 81.0
Lets look at a scatter plot. Linear,
exponential or other?
76.3
56.7
75.3
70.1
65.9
61.9
Lets look at a scatter plot. Linear,
exponential or other?
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How to Find the best fit exponential function
through the data

• Enter the data into L1 and L2.
• Go to STAT ? CALC
• Go down to or hit 0 ExpReg and enter
• Enter this equation in the Y and view how well
  it fits the data

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• From your model, by what percentage do the tire
  sales grow each year?
• What year (correct to 3 decimal places) did the
  company first sell 450 tires?
• If the growth continued at this rate, how many
  tires would the company have sold in 1998?How
  does this value compare with the last value in
  the table.