Question

How do you find the average rate of change for the function `y=X^2 + 5X` on the indicated intervals [2,5]?

Answer 1

The average rate of change is 2

Explanation:

In general the rate of change of `y=x^2+5x` is
`color(white)("XXXX")(dy)/(dx)=2x+5`

At `x=5`
`color(white)("XXXX")(dy)/(dx) = 2(5)+5 = 15`

At `x=2`
`color(white)("XXXX")(dy)/(dx)=2(2)+5=9`

Between `x=2` and `x=5`
`color(white)("XXXX")(dy)/(dx)` changes by `15-9=6`

The distance between (`x=2`) and (`x=5`) is `5-2 = 3`

The average rate of change (per unit change in `x`) is
`color(white)("XXXX")6/3 =2`

Answer 2

The average rate of change of a function `f` on an interval `[a,b]` is: `(Deltaf)/(Deltax) = (f(b)-f(a))/(b-a)`.

Explanation:

The average rate of change of a function `f(x) = x^2+fx` on an interval `[2,5]` is:

`(Deltaf)/(Deltax) = (f(5)-f(2))/((5)-(2))`

`= (50 - 14)/(3) = 36/3 = 12`