Question

find the average rate of change of f(x) = 3x^2 + 1
(a) from 2 to 4
(b) from -2 to 0
and
(c) from 2 to 5

Answer 1

Average rate of change is: \[
\frac{f(x_2)-f(x_1)}{x_2-x_1}
\]I could derive it for you if you're curious.

Answer 2

yes please

Answer 3

Well we know that the rate of change is \(f'(x)\).
So we want to find the average of that function.
The average of a function is give by \[
g_{average} = \frac{1}{x_2-x_1}\int_{x_1}^{x^2}g(x)dx
\]

Answer 4

So we let \(g = f'\) and get: \[
f'_{average} = \frac{1}{x_2-x_1}\int_{x_1}^{x_2}f'(x)dx = \frac{f(x_2)-f(x_1)}{x_2-x_1}
\]I just used the fundamental theorem of Calculus on that second step.

Answer 5

Ok so how would I plug in the equation with the rate of change equation?

Answer 6

I always mess that part up for some reason....

Answer 7

(a) from 2 to 4

This means \(x_1 = 2,x_2=4\).

So they want \[
\frac{f(4)-f(2)}{4-2} = \frac{(3(4)^2 + 1)-(3(2)^2 + 1)}{4-2}
\]

Answer 8

omg that is a lot

Answer 9

So your replacing x for 4 and 2?

Answer 10

Yeah...

Answer 11

So how would I get the rate of change?