Question

Find the average rate of change of each function on the interval specified in simplest form. Do not type your answer in factored form and do not type any spaces between characters. g(x)=4x^2-7 on the interval [1,b]The average rate of change is Answer

Answer 1

The average rate of change is: 4(b + 1)

How to find the average rate of change?

The formula for the average rate of change between the interval (a, b) is:

f'(x) =
`(f(b) - f(a))/((b - a))`

The function is given as:

g(x) = 4x² - 7

Thus, the procedure to find the average rate of change on the interval [1,b] is:

g(1) = 4(1)² - 7

g(1) = -3

g(b) = 4(b)² - 7

g(b) = 4b² - 7

g'(x) = (4b² - 7 - (-3))/(b - 1)

g'(x) = (4b² - 4)/(b - 1)

g'(x) = 4(b + 1)(b - 1)/(b - 1)

g'(x) = 4(b + 1)

Answer 2

The average rate of funtion is defined as :

`(f(b)-f(a))/(b-a)\to\mleft\lbrack a,b\mright\rbrack`

From the problem we have :

`g(x)=4x^2-7\to\mleft\lbrack1,b\mright\rbrack`

We evaluate at each of the extremes of the interval

`\begin{gathered} g(1)=4\cdot(1)^2-7 \\ g(1)=4-7 \\ g(1)=-3 \end{gathered}`
`g(b)=4b^2-7`

We replace in the equation the average rate of change

`\begin{gathered} ((4b^2-7)-(-3))/(b-1)=(4b^2-4)/(b-1)=(4(b^2-1))/(b-1) \\ \end{gathered}`

factoring

`b^2-1=(b-1)(b+1)`

We simplify the average rate of change

`((4b^2-7)-(-3))/(b-1)=\frac{4(b^{}-1)(b+1)}{b-1}=4(b+1)`

Ethnoces the answer is 4(b+1)