Question

Which expression below gives the average rate of change of the function h(x)=4^(x+2)+7 on the interval –3 ≤ x ≤ 5?

Answer 1

`(4^8-1)/(32)`

Explanation:

`h(x)=4^((x+2)) +7`

We know that formula for average rate of change of a function from a to b is

`(f(b)-f(a))/(b-a)`

Here f equals h and a =-3 b=5

h(a) =
`4^(5+2) +7 = 4^7+7`

and

h(b) =
`4^(-3+2) +7 = (1)/(4) +7`

h(b)-h(a) =
`4^7-(1)/(4) =(4^8-1)/(4)`

b-a = 5-(-3) =8

Hence average rate of change is

`(4^8-1)/(4)((1)/(8) )=(4^8-1)/(32)`

Answer 2

Answer: 2048

Explanation:

h(x) = 4ˣ⁺² + 7

h(5) = 4⁵⁺² + 7

= 4⁷ + 7

h(-3) = 4⁻³⁺² + 7

= 4⁻¹ + 7

Average rate of change is the slope of the interval.

`(y_2-y_1)/(x_2-x_1) = (h(5)-h(-3))/(5 - (-3))=((4^7+7)-(4^(-1)+7))/(5 + 3) = (16391-7.25)/(8)=2048`