Question

25 POINTS answer question 4 pleaseeeee

Answer 1

Answer:

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

Step-by-step explanation:

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

We need to find Average rate of change

The formula used is: $Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}$

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.

$Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}\\Average \ rate \ of \ change=\frac{16-8}{2-1} \\Average \ rate \ of \ change=\frac{8}{1}\\Average \ rate \ of \ change=8$

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.

$Average \ rate \ of \ change=\frac{f(b)-f(a)}{b-a}\\Average \ rate \ of \ change=\frac{64-32}{4-3} \\Average \ rate \ of \ change=\frac{32}{1}\\Average \ rate \ of \ change=32$

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.