What is the average rate of change of h(x)=2^{x+1}h(x)=2 x+1 h, (, x, ), equals, 2, start superscript, x, plus, 1, end superscript over the interval [2,4][2,4]open bracket, 2, comma, 4, close bracket?

Average Rate = 12

Explanation:

Given

h(x) = 2^(x + 1)

Interval = [2,4]

Required

Determine the average rate of change

The average rate of change of h(x) is calculated using.

Average Rate = (h(b) - h(a))/(b - a)

Where

[a,b] = [2,4]

Meaning a = 2 and b = 4

So, the formula becomes:

Average Rate = (h(4) - h(2))/(4 - 2)

Average Rate = (h(4) - h(2))/2

Average Rate = ½(h(4) - h(2))

Calculating h(4)

h(4) = 2^(4+1)

h(4) = 2⁵

h(4) = 32

Calculating h(2)

h(2) = 2^(2+1)

h(2) = 2³

h(2) = 8

So, we have:

Average Rate = ½(h(4) - h(2))

Average Rate = ½(32 - 8)

Average Rate = ½ * 24

Average Rate = 12

Hence, the average rate of change is 12