Question

How do you find the average rate of change of `f(x)=3x-2x^2` over interval [-3,4]?

Answer 1

So mean rate of change is `(15-13)/2= +1`

Explanation:

As you use the term `f(x)` I am assuming you are using early stage Calculus.

Set `y=3x-2x^2` ........................(1)

Increment `x` by the minute amount of `deltax`
The this will cause a minute change in y of `deltay`

So

`y+deltay=3(x+deltax)-2(x+deltax)^2`

`=> y+deltay=3x+3deltax-2(x^2+2xdeltax+(deltax)^2)`

`=> y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2)`

`=> y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2`....(2)

Subtract equation (1) from equation (2)

`" " y+deltay=3x+3deltax-2x^2-4xdeltax-2(deltax)^2`....(2)
`" "underline( y" "= 3x" " -2x^2color(white)(........................) -) ..(1)`
`" " deltay=0" "+3deltax+0-4xdeltax-2(deltax)^2`

Divide throughout by `deltax`

`(deltay)/(deltax)= 3-4x-2deltax`

`lim_(deltaato0)(deltay)/(deltax)= 3-4x- lim_(deltaxto0)(2deltax)`

`(dy)/(dx)=3-4x`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
At `x=-3" "->" " (dy)/(dx)=3-4(-3) = +15`

At `x=+4" "->" " (dy)/(dx)=3-4(+4) = -13`

So mean rate of change is `(15-13)/2= +1`