Answer:
a) The average rate of change for the function
over interval x=1 to x=2 is 8
b) The average rate of change for the function
over interval x=3 to x=4 is 32
Step-by-step explanation:
We are given function ![f(x)=4(2)^x](https://tex.z-dn.net/?f=f%28x%29%3D4%282%29%5Ex)
We need to find Average rate of change
The formula used is: ![Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}](https://tex.z-dn.net/?f=Average%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bb-a%7D)
Part a) The interval is: x=1 to x=2
We have a= 1 and b=2
Finding f(a) i.e f(1)
![f(x)=4(2)^x\\Put \ x =1\\f(1)=4(2)^1\\f(1)=4(2)\\f(1)=8](https://tex.z-dn.net/?f=f%28x%29%3D4%282%29%5Ex%5C%5CPut%20%5C%20x%20%3D1%5C%5Cf%281%29%3D4%282%29%5E1%5C%5Cf%281%29%3D4%282%29%5C%5Cf%281%29%3D8)
Now finding f(b) i.e f(2)
![f(x)=4(2)^x\\Put \ x =2\\f(2)=4(2)^2\\f(2)=4(4)\\f(2)=16](https://tex.z-dn.net/?f=f%28x%29%3D4%282%29%5Ex%5C%5CPut%20%5C%20x%20%3D2%5C%5Cf%282%29%3D4%282%29%5E2%5C%5Cf%282%29%3D4%284%29%5C%5Cf%282%29%3D16)
Now finding average rate of change.
![Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}\\Average \ rate \ of \ change=\frac{16-8}{2-1} \\Average \ rate \ of \ change=\frac{8}{1}\\Average \ rate \ of \ change=8](https://tex.z-dn.net/?f=Average%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bb-a%7D%5C%5CAverage%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7B16-8%7D%7B2-1%7D%20%5C%5CAverage%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7B8%7D%7B1%7D%5C%5CAverage%20%5C%20rate%20%5C%20of%20%5C%20change%3D8)
So, average rate of change for the function
over interval x=1 to x=2 is 8
Part b)
The interval is: x=3 to x=4
We have a= 3 and b=24
Finding f(a) i.e f(3)
![f(x)=4(2)^x\\Put \ x =3\\f(3)=4(2)^3\\f(3)=4(8)\\f(3)=32](https://tex.z-dn.net/?f=f%28x%29%3D4%282%29%5Ex%5C%5CPut%20%5C%20x%20%3D3%5C%5Cf%283%29%3D4%282%29%5E3%5C%5Cf%283%29%3D4%288%29%5C%5Cf%283%29%3D32)
Now finding f(b) i.e f(4)
![f(x)=4(2)^x\\Put \ x =4\\f(4)=4(2)^4\\f(4)=4(16)\\f(4)=64](https://tex.z-dn.net/?f=f%28x%29%3D4%282%29%5Ex%5C%5CPut%20%5C%20x%20%3D4%5C%5Cf%284%29%3D4%282%29%5E4%5C%5Cf%284%29%3D4%2816%29%5C%5Cf%284%29%3D64)
Now finding average rate of change.
![Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}\\Average \ rate \ of \ change=\frac{64-32}{4-3} \\Average \ rate \ of \ change=\frac{32}{1}\\Average \ rate \ of \ change=32](https://tex.z-dn.net/?f=Average%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7Bf%28b%29-f%28a%29%7D%7Bb-a%7D%5C%5CAverage%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7B64-32%7D%7B4-3%7D%20%5C%5CAverage%20%5C%20rate%20%5C%20of%20%5C%20change%3D%5Cfrac%7B32%7D%7B1%7D%5C%5CAverage%20%5C%20rate%20%5C%20of%20%5C%20change%3D32)
So, average rate of change for the function
over interval x=3 to x=4 is 32
Since the interval is increasing, so does our average rate of change increasing as function is also increasing.