Walleye
Prove that if f(x) = integral from 0 to x of f(t) dt then f = 0



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Walleye
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if \[f(x) = \int\limits_{0}^{x} f(t) dt \]
then f = 0

Mr.Math
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This is not a true statement.

Walleye
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Michael Spivak claims it is

Mr.Math
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Who's Michael Spivak?

Walleye
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The man who wrote my textbook.

Mr.Math
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Whoever he might be, tell him Newton has another opinion :P

Mr.Math
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This is a direct use of the fundamental theorem of calculus.

imranmeah91
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\[\int_0^x 2t dt= t^2 \]
\[x^2\]

Mr.Math
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Oh wait!

Walleye
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haha im not going anywhere with this one

across
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It implies f(0)=0.

Walleye
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hes claiming f(x) = 0 for any x

Walleye
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I certinally dont see it

imranmeah91
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is there derivative sign infront of integral?

Walleye
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Nope that is the whole question

Mr.Math
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I didn't read the question well at first. This means that f is an antiderivative of itself, if I'm seeing this right.

Walleye
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yes f'(x) = f(x)

across
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We know that\[f(x)=\int f(x)dx\implies f(x)=e^x\]^^

Mr.Math
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Yeah.

Walleye
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oh nice

Walleye
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I didnt think about e^x with this one

Mr.Math
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Then differentiate both sides you get, f'(x)=f(x), which is an ODE that has the solution \(f(x)=ce^{x}\).

Mr.Math
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Your statement is still not correct :P

Walleye
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Welllll hold on

Zarkon
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it is correct....find c

Walleye
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the e^x makes senese with f'(x) = f(x) but this function is an integral

Mr.Math
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You're saying c=0 @Zarkon.

Zarkon
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\[f(0) = \int\limits_{0}^{0} f(t) dt=0\]

Walleye
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It wants me to prove the function is 0

Zarkon
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\[\Rightarrow c=0\]

Mr.Math
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That's right! I'm a loser!! :(

Walleye
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No you're not!
So basically because f'(x) = f(x) I can say f(x) = ce^x and then show that f(0) = 0 implying that c=0 so f = 0

Mr.Math
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Exactly!

Walleye
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makes perfect sense!
Im just be a loser now but how do we know there is no other function s.t. f'(x) = f(x)

across
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Mr. Michale Spivak did a good job, that is, to elicit eager students to congregate and think this one through. xd

Walleye
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I hate michale spivak :P

Walleye
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I'm just kidding its just a challening course for me

Mr.Math
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Lol @across.
@Wall, this is a first order homogeneous equation and had only this solution, \(i.e f(x)=ce^{x}\).

Mr.Math
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homogeneous differential equation* and it has* *_*

Walleye
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ahhh yes! I really like this problem! I can't believe I didnt realize f(x) had to be some form of e^x Thanks for all the help everyone!!!!

Mr.Math
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You're welcome! Thanks for fanning me :D