For the exponential function represented in the table below

For the exponential function represented in the table below, determine the initial value, the n -unit growth factors, and the exponential function that models the data.

The initial value:
   

The 1-unit growth factor:
   

The 1-unit percent change:
%   

The 3-unit growth factor:
   

The 1/4-unit growth factor:
   

Function definition:

Solution

Solution:let exponential function=Ke^-nx (negetive sign for decresing function)

put x=1   f(x)=Ke^-nx ,then 739.8=Ke^{-n                          .........................}1]

put x=2            then 539.31=Ke^{-4n                                      .....................}2]

dividing both eq. and solving thies eq. we get K=821.78 & n=.1051

then the function is f(x)=821.78e^-.1051x

1]: the initial value at x=0 is =821.78

2]: the chenge in decaying =(f(1)-f(2))/f(1)

                      =.0997

growth fector is=1-.0997=.9003                        (for subtrecting from 1, becouse function is decaying)

3]: one unit percantege chenge=(f(1)-f(2))/f(1)*100

          =(1-e^-.1051)*100

         =0.0997*100=9.997%

4]: the 3 unit growth unit fector is=(f(1)-f(4))/f(1)

=1-e^-3*.1051=.9947

=1-.9980=.005232

5]: the 1/4 unit growth fector is=1-f(5/4)/f(1)

                                          =.0259

for decaying function =1-.0259=.9740

6]: defination: this is exponential decaying function