Let gx int0x ft dt where f is the function whose graph is...

Let $g(x) = \int_0^x f(t) \, dt$, where $f$ is the function whose graph is shown. (a) Evaluate $g(0)$, $g(5)$, $g(10)$, $g(15)$, and $g(30)$. $g(0) = 0$ $g(5) = 50$ $g(10) = 125$ $g(15) = 175$ $g(30) = 275$ (b) On what interval is $g$ increasing? (Enter your answer using interval notation.) $(0, 15)$ (c) Where does $g$ have a maximum value? $x = 15$

Transcript

00:01 Alright, so here we are given g of x equal to the integral from zero to x of f of t dt, where f is the function that's shown here.

00:09 First we want to evaluate some of our numbers here.

00:12 So g of zero is going to equal the integral from zero to zero of f dt.

00:20 So for integrating from zero to zero, then that is going to be a value of zero, alright? g of five is then going to be the integral from zero to five of f dt, alright? so that's going to be this integral right here.

00:39 Alright, so from zero to five, that's a rectangle, so we're going to take five and multiply it by the height.

00:45 Alright, so each of the squares have a value of five, so that means there's two squares here, which means it has a length of five and a width of 10.

00:56 So the area is five times 10, which is going to be 50.

01:01 Alright, so then for g of 10, it's going to be now the integral from zero to 10.

01:07 Alright, so that's going to be including this whole entire piece right here.

01:12 Alright, so right away we see we have this rectangle here.

01:16 Alright, that's just two of the integrals from zero to five.

01:20 So we know we're going to have, this is equal to 100 plus this triangle that's given.

01:27 So we have half of one of the rectangles that we found, so it's going to be half of 50, which is 25, which means our area is 125.

01:38 Alright, then for g of 15, that's the integral from zero to 15 of f.

01:48 Alright, so that is this area we just found, so 125 plus this area right here.

01:58 Alright, so we have a base of five and then a height of five, 10, 15, 20.

02:06 Alright, so we're going to take half of five and 20, alright, which that is then going to be equal 125, and i'll write this actually below here.

02:20 It'll be 125 plus 50, which is going to equal 175.

02:28 Then we have g of 30...

Question

Please give Ace some feedback