Question

What is the average rate of change of the function f (x) =2x^2+8 over the interval [2, 6]?241216

Answer 1

Given:

`f(x)=2x^2+8`
`[2,6]`

To determine the average rate of change, we use the formula:

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

where:

[a,b] = interval

A(x)= average rate of change

Based on the given formula, we let a=2 and b=6. Hence,

`\begin{gathered} A(x)=(f(b)-f(a))/(b-a) \\ A(x)=(f(6)-f(2))/(6-2) \end{gathered}`

We plug in x=2 into f(x)=2x^2+8:

`\begin{gathered} f(x)=2x^(2)+8 \\ f(2)=2(2)^2+8 \\ Simplify \\ f(2)=16 \end{gathered}`

We plug in x=6 into f(x)=2x^2+8:

`\begin{gathered} f(x)=2x^(2)+8 \\ f(6)=2(6)^2+8 \\ f(6)=80 \end{gathered}`

So,

`\begin{gathered} A(x)=(f(6)-f(2))/(6-2) \\ A(x)=(80-16)/(6-2) \\ Simplify \\ A(x)=16 \end{gathered}`

Therefore, the average rate of change is 16.