Fundamental Theorem Calc: Find f(4) from Integral

[Jet1045]

Alright, so in my AP calc class we just got a worksheet and one of the questions i don't undersnat at allll! We have been learning about the Fundamental Theorem of Calculus recently, so I am guessing that is what this is about.

Homework Statement

Find f(4) if the integral (lower limit = 0 , upper limit = x) f(t) dt = xcos(pi(x))2. The attempt at a solution

Honestly I don't even know where to start. If you are given an integral, how do you get the original equation f(t) to even insert 4 into?

Sorry if the question is unclear, I am unsure how to actually type out integrals and such on this forum, if someone can give me a link explaining how to , i can rewrite the question so it is easier to understand.

Thanks :)

 

[sutupidmath]

Take the derivative of both sides!

 

[Mark44]

Click on the equation below to see the LaTeX I used.
\int_0^x f(t)dt = x cos(\pi x)

 

[Jet1045]

Thank Mark! That will make it a lot easier to ask questions in the future :)

and surupidmath, you mean to take the derivative of xcos(pi(x))?
If I do I get

<br /> cos(\pi x) - xsin(\pi x)\pi<br />

can i unsert 4 into that now?

 

[sutupidmath]

What does the first part of the FTOC say? look it up.

 

[Jet1045]

Well we just started learning about the fundamental theorem, so therefore my knowledge on the subject is pretty limited. Looking it up online will most likely just confuse me more hence why I am asking questions here.

All i need to know is if by taking the derivative of xcos(pi(x)) does that give the original function , f(t), for which i can put 4 into.

 

[Mark44]

If you differentiate both sides of this equation:
\int_0^x f(t)dt = x cos(\pi x)

you get
cos(\pi x) - \pi xsin(\pi x)
on the right side.

What do you get on the left side of this equation?

I'm assuming you have a textbook that talks about the FTC. See what it says there.

 

[HallsofIvy]

Well we just started learning about the fundamental theorem, so therefore my knowledge on the subject is pretty limited.[/qutoe]
Usually, the first thing you see on learning about something is a statement of what it is!

Are you saying you have not yet seen a statement of what the "Fundamental Theorem of Calculus" says?

Looking it up online will most likely just confuse me more hence why I am asking questions here.

All i need to know is if by taking the derivative of xcos(pi(x)) does that give the original function , f(t), for which i can put 4 into.

The "Fundamental Theorem of Calculus" has two parts:
1) If F(x) is a differentiable function then its derivative is integrable and
\int_a^b \frac{dF}{dx} dx= F(b)- F(a).

2) If f(x) is an integrable function then its integral
F(x)= \int_a^x f(t)dt
is differentiable and
f(x)= \frac{dF}{dx}


Essentially it says that differentiation and integration are "inverse" operations.