Question

Find the average rate of change for f(x) = 6x2 + 3 over the interval [x, x + h).

Answer 1

Find the average rate of change for f(x) = 6x2 + 3 over the interval [x, x + h).

The average rate of change is equal to

`(f(x+h)-f(x))/(h)`

so

f(x)=6x^2+3

f(x+h)=6(x+h)^2+3=6(x^2+2hx+h^2)+3=6x^2+12hx+6h^2+3

substitute the given values

`(6x^2+12hx+6h^2+3-(6x^2+3))/(h)`
`(6x^2+12hx+6h^2+3-6x^2-3)/(h)`

simplify

`(12hx+6h^2)/(h)=12x+6h`

the answer is 12x+h