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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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Chapter
Section
BuyFindarrow_forward

Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 11-16, choose the logistic function that best approximates the given curve.

images

(A) f ( x ) = 10 1 + 3 ( 1.01 ) x

(B) f ( x ) = 8 1 + 7 ( 0.1 ) x

(C) f ( x ) = 10 1 + 3 ( 0.1 ) x

To determine

The correct logistic function f

from the given choices for the following curve,

images

(A). f(x)=101+3(1.01x)

(B). f(x)=81+7(0.1x)

(C). f(x)=101+3(0.1x)

Explanation

Given Information:

The provided curve is,

The given choices for the correct logistic function are,

(A). f(x)=101+3(1.01x)

(B). f(x)=81+7(0.1x)

(C). f(x)=101+3(0.1x)

The form of the logistic function is:

f(x)=N1+A(bx)

Where constants N,A

and b

are non-zero (A,bare positive and b1).

There are certain properties of the curve represented by the logistic curve, y=N1+A(bx)

1.

The graph of the function formed is S-shaped which is formed by sandwiching between the horizontal lines y=0 and y=N, which means the limiting value of the function is N.

2.

If b>1, graph rises, if b<1, the graph falls.

Now, the provided curve is,

It is clear from the curve that the function f

has limiting value 10, which means the value of constant N=10.

Now, consider the general logistic equation,

f(x)=N1+A(bx)

The graph illustrates that the given curve passes through the point, (0,2.5)

Substitute 10

for N, 0

for x and 2.5

for f(x) in f(x)=N1+A(bx):

2.5=101+A(b(0))2.5=101+A

Multiply both sides by (1+A):

2.5(1+A)=101+A(1+A)2

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