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Section 3.5

Modeling with ExponentialLogarithmic Functions

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Exponential Growth and Decay

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Example

0The equation A=A models growth of a deerpopulation in a small local preserve. If the initial population

is 14 deer and the population grows to 20 in 4 years,

find the value of k, the rate of growt

kt

e

h.

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Logistic Growth Models

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Example

If the following logistic equation models the number of people

who become infected with noro virus on a small cruise ship off

the coast of Alaska, how many people will become ill by the second

week if t

1.5

is given in weeks.

500f(t)=

1

t

e

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The Art of Modeling

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Graphing Calculator - Exponential Regression

Press STAT then EDIT 1. Type in the values in List 1 (x values)and List 2 (y values).

Then STAT, move over to CALC and press 0. You will get valuesfor a, b, rand r2. The values of aand bgo into the equationy=a*b^x or y=abx.

When you use a graphing utility the value of MAY appear.

This is called the correlation coefficient and is a measure of

how well the model fits the data. The value of is such that

-1 1. A pos

r

r

r

r

itive means that as the x-values increase, sodo the y-values. A negative means that as the x-values increase,

the y values decrease. The closer that is to -1 or 1, the better the

model fits the

rr

r

data.

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Example

L1 L2

1 8

2 18

3 49

4 123

5 312

6 780

Find the exponential regression equation on your calculator usingthe data below. The data is for the population of black flies

which increases exponentially in a favorable lab environment.

L1 is the number of weeks and L2 is for the number of black

flies in the laboratory experiment.

See the next screen for continuation of example.

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Example-Graphing Calculator

Continuing with this same data, we are now going to graph the equationthat we got earlier in this exercise and we are going to graph the data.

First make certain that your window is large enough so that you will

be able to see the graphed points.

Second, go to STAT PLOT and press ENTER on #1. Press ENTER

on the word "On", for type of graph use the first one, make Xlist: L1,and Ylist:L2. Use either mark for the last option.

Now return to the graph and see the data.

Go to Y= and type in the equation that you got on the previous screen.

Return to the graph and observe how close the curve is to the actual data.

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How can we obtain a logarithmic function that models

the data? A graphing utility can be used to obtain a logarithmic

model of the form y=a+b ln x. Because the domain of the

logarithmic function is the set of positive numbers, zero must

not be a value for x.

The technique for finding that logarithmic equation is the same

as for finding the exponential equation on a graphing calculator.

On the STAT, CALC menu choose 9 for Logarithmic Regression.

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Expressing y=abx in Base e

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Example

Rewrite y=3.82 (6.7) in terms of base e.x

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(a)

(b)

(c)

(d)

0A=A models growth of gray squirrels in a small city park.

If the initial population is 30 in 2000, and it grows to 38 in 1 year,

what is the growth rate, k?

kte

.236

.023

.135

.126

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(a)

(b)

(c)

(d)

1000The equation f(x)= models a rare type of flu

1 80

which is infecting a town school. How many students, f(x),will be sick in 2 weeks if is time in weeks?

te

t

102

36

85

94