Question

F(x)= -2x^2+3x-7, calculate the average rate of change on the interval [1, 1+h]

Answer 1

`Rate = -2h-1`

Explanation:

Given

`f(x) = -2x^2 + 3x - 7`

`Interval: [1,1+h]`

Required

The average rate of change

This is calculated as:

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

Where:

`[a,b] = [1,1+h]`

So, we have:

`Rate = (f(1+h) - f(1))/(1+h - 1)`

`Rate = (f(1+h) - f(1))/(h)`

Calculate f(1+h) and f(1)

`f(x) = -2x^2 + 3x - 7`

`f(1) = -2 * 1^2 + 3 * 1 - 7 = -6`

`f(1+h) = -2 * (1+h)^2 + 3 * (1+h) - 7`

Evaluate squares and open bracket

`f(1+h) = -2 * (1+h+h+h^2) + 3+3h - 7`

`f(1+h) = -2-2h-2h-2h^2 + 3+3h - 7`

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

Collect like terms

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

`f(1+h) = -2h^2-h-6`

So, we have:

`Rate = (f(1+h) - f(1))/(h)`

`Rate = (-2h^2-h-6 - -6)/(h)`

`Rate = (-2h^2-h-6 +6)/(h)`

`Rate = (-2h^2-h)/(h)`

Factorize the numerator

`Rate = (h(-2h-1))/(h)`

Divide

`Rate = -2h-1`