Question

The fundamental theorem of Calculus. Help?

Answer 1

@zepdrix @phi @dan815

Answer 2

Ooo the 4th one is pretty fun :)

Answer 3

ok, so you want to integrate y over the interval, can you write the experssion

Answer 4

Is my 1st one right?

Answer 5

yes

Answer 6

For number 2 I got 1/3 is that right too?

Answer 7

yes c:

Answer 8

ok so number 3 and 4 I am confused

Answer 9

I know for 4 they are looking for the inverse but how do i get it?

Answer 10

Inverse? :o
Derivative silly! :)

Answer 11

ohhhhhhh my bad nvm give me a second

Answer 12

You have to apply the Fundamental Theorem of Calculus, Part 1:\[\large\rm \frac{d}{dx}\int\limits_c^x g(t)dt=g(x)\]
Actually it might be useful to use BOTH parts of the theorem for this problem.

Answer 13

I got 18x^3 for the derivative of it.

Answer 14

but that isnt a choice =C

Answer 15

For nuber 3 I got B. and number 4 I don't know.

Answer 16

I got 18x^3 when I integrated but 0 for the derivative...**

Answer 17

for 4 you have to experss F as a function of x, not as intergration

Answer 18

I got 18x^3 when i integrated bu 0 as the derivative**

Answer 19

This is what we're trying to figure out problem 4,\[\large\rm \frac{d}{dx}\int\limits_{-3x}^{3x}s^2ds\]Try to think of it more generally like this,\[\large\rm \frac{d}{dx}\int\limits_{-3x}^{3x}f(s)ds\]You anti-differentiate to get some function, let's call it F(s)

Answer 20

\[\large\rm \frac{d}{dx}\int\limits\limits_{-3x}^{3x}f(s)ds\quad=\frac{d}{dx}F(s)|_{-3x}^{3x}\]

Answer 21

Plug in your limits,\[\large\rm \frac{d}{dx}\left[F(3x)-F(-3x)\right]\]and now take a derivative

Answer 22

Big F turns back into the original function, little f,
but we have some chain rule going on, right?\[\large\rm =3f(3x)--3f(-3x)\]

Answer 23

yeah

Answer 24

little f is the thing we started with, s^2 in this case,
So now we have the 3x and -3x being squared, replacing the s value,\[\large\rm =3(3x)^2+3(-3x)^2\]And simplify! :)

Answer 25

It's a weird little trick.
If you got confused somewhere in that process, I wouldn't be surprised.

Answer 26

54x^2 so c. Now number 3.

Answer 27

so b*

Answer 28

I'm going to close this one and open a new one