To find the average rate of change over the given intervals, we need to remember that the average rate of change is given by:
[tex]A_{\text{rateofchange}}=\frac{change\text{ in y}}{\text{change in x}}_{}=\frac{f(b)-f(a)}{b-a}[/tex]
We already have that the function is:
[tex]f(x)=55(0.7)^x[/tex]
Then, if we want to find the average rate of change over the interval:
[1,3] (expressed in interval notation), we have to use the values for x = 1, and x = 3 as inputs for the function. Then, we have:
the average rate of change over
[tex]A_{\text{rateofchange}}=\frac{f(3)-f(1)}{3-1}[/tex]
And we have that:
[tex]f(x)=55(0.7)^x\Rightarrow f(3)=18.865[/tex]
[tex]f(1)=38.5[/tex]
Then the rate of change in the given interval is:
[tex]A_{\text{rateofchange}}=\frac{18.865-38.5}{3-1}=\frac{-19.635}{2}=-9.8175[/tex]
If we round the result to three decimal places, we have that the average rate of change in the interval is -9.818.
We can follow the same procedure for the interval [4, 8] as follows:
[tex]f(x)=55(0.7)^x\Rightarrow f(8)=55(0.7)^8_{}=3.17064055[/tex]
[tex]f(4)=55(0.7)^4\Rightarrow f(4)=13.2055[/tex]
If we apply the concept of the average rate of change, then we have:
[tex]A_{\text{rateofchange}}=\frac{f(8)-f(4)}{8-4}=\frac{3.17064055-13.2055}{8-4}[/tex]
Then, we have:
[tex]A_{\text{rateofchange}}=\frac{-10.03485945}{4}\Rightarrow A_{rateofchange}=-2.5087148625[/tex]
If we round the result to three decimal places, then we have that the average rate of change over the interval [4, 8] is -2.509.
Since we have that over the interval [1, 3] is -9.818, and over the interval [4, 8] the average rate of change is -2.509, we can conclude that it increases as x increases.
