Question

What is the average rate of change of the function f(x)=-3x^2 + 7x+ 15 over the interval -2 ≤ x ≤ 2 ?

Please explain how to solve it!
Thank you

Answer 1

Explanation:

The average rate of change of the function is represented by the following:

`Average\ rate\ of\ change\ =(Change\ in\ y\ )/(Change\ in\ x\ )`

When we think about average rate of change of a function we actually need to calculate the slope of the line between the interval on the function in this case the function is f(x) = -3x^2 + 7x + 15 over the interval -2 ≤ x ≤ 2

in this case y = f(x) so here average rate of change is,

Average rate of change = Δf / Δx

so to calculate Δf we use the closed interval that is given
`-2\leq x\leq 2 \\`

so here goes,

`f(x)=-3x^2+7x+15\\f(2)=-3(2)^2+7(2)+15\\f(2)=-12+14+15\\f(2)=17\\\\f(x)=-3x^2+7x+15\\f(-2)=-3(-2)^2+7(-2)+15\\f(-2)=-12-14+15\\f(-2)=-11\\`

so now , Δf = f(2) - f(-2) = final value of f(x) - initial value of f(x)

Δf = 17 - 11

Δf = 6 = Change in y

now we need Change in x which means Δx

so now,

Δx = final value of x - initial value of x

Δx = 2 - ( -2 )

Δx = 4

so now the Average rate of change = Δf / Δx

Average rate of change = 6/4

Average rate of change = 3/2

I have attached an image for you to visualize it clearly