Question

25 POINTS
answer question 4 pleaseeeee

Answer 1

a) The average rate of change for the function
`f(x)=4(2)^x` over interval x=1 to x=2 is 8

b) The average rate of change for the function
`f(x)=4(2)^x` over interval x=3 to x=4 is 32

Explanation:

We are given function
`f(x)=4(2)^x`

We need to find Average rate of change

The formula used is:
`Average \ rate \ of \ change=(f(b)-f(a))/(b-a)`

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`

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`

Now finding average rate of change.

`Average \ rate \ of \ change=(f(b)-f(a))/(b-a)\\Average \ rate \ of \ change=(16-8)/(2-1) \\Average \ rate \ of \ change=(8)/(1)\\Average \ rate \ of \ change=8`

So, average rate of change for the function
`f(x)=4(2)^x` 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`

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`

Now finding average rate of change.

`Average \ rate \ of \ change=(f(b)-f(a))/(b-a)\\Average \ rate \ of \ change=(64-32)/(4-3) \\Average \ rate \ of \ change=(32)/(1)\\Average \ rate \ of \ change=32`

So, average rate of change for the function
`f(x)=4(2)^x` 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.