Answer:
a) The average rate of change for the function [tex]f(x)=4(2)^x[/tex] over interval x=1 to x=2 is 8
b) The average rate of change for the function [tex]f(x)=4(2)^x[/tex] over interval x=3 to x=4 is 32
Step-by-step explanation:
We are given function [tex]f(x)=4(2)^x[/tex]
We need to find Average rate of change
The formula used is: [tex]Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}[/tex]
Part a) The interval is: x=1 to x=2
We have a= 1 and b=2
Finding f(a) i.e f(1)
[tex]f(x)=4(2)^x\\Put \ x =1\\f(1)=4(2)^1\\f(1)=4(2)\\f(1)=8[/tex]
Now finding f(b) i.e f(2)
[tex]f(x)=4(2)^x\\Put \ x =2\\f(2)=4(2)^2\\f(2)=4(4)\\f(2)=16[/tex]
Now finding average rate of change.
[tex]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[/tex]
So, average rate of change for the function [tex]f(x)=4(2)^x[/tex] 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)
[tex]f(x)=4(2)^x\\Put \ x =3\\f(3)=4(2)^3\\f(3)=4(8)\\f(3)=32[/tex]
Now finding f(b) i.e f(4)
[tex]f(x)=4(2)^x\\Put \ x =4\\f(4)=4(2)^4\\f(4)=4(16)\\f(4)=64[/tex]
Now finding average rate of change.
[tex]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[/tex]
So, average rate of change for the function [tex]f(x)=4(2)^x[/tex] 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.
