anonymous
  • anonymous
Prove that if f(x) = integral from 0 to x of f(t) dt then f = 0
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
if \[f(x) = \int\limits_{0}^{x} f(t) dt \] then f = 0
Mr.Math
  • Mr.Math
This is not a true statement.
anonymous
  • anonymous
Michael Spivak claims it is

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Mr.Math
  • Mr.Math
Who's Michael Spivak?
anonymous
  • anonymous
The man who wrote my textbook.
Mr.Math
  • Mr.Math
Whoever he might be, tell him Newton has another opinion :P
Mr.Math
  • Mr.Math
This is a direct use of the fundamental theorem of calculus.
anonymous
  • anonymous
\[\int_0^x 2t dt= t^2 \] \[x^2\]
Mr.Math
  • Mr.Math
Oh wait!
anonymous
  • anonymous
haha im not going anywhere with this one
across
  • across
It implies f(0)=0.
anonymous
  • anonymous
hes claiming f(x) = 0 for any x
anonymous
  • anonymous
I certinally dont see it
anonymous
  • anonymous
is there derivative sign infront of integral?
anonymous
  • anonymous
Nope that is the whole question
Mr.Math
  • Mr.Math
I didn't read the question well at first. This means that f is an anti-derivative of itself, if I'm seeing this right.
anonymous
  • anonymous
yes f'(x) = f(x)
across
  • across
We know that\[f(x)=\int f(x)dx\implies f(x)=e^x\]^^
Mr.Math
  • Mr.Math
Yeah.
anonymous
  • anonymous
oh nice
anonymous
  • anonymous
I didnt think about e^x with this one
Mr.Math
  • Mr.Math
Then differentiate both sides you get, f'(x)=f(x), which is an ODE that has the solution \(f(x)=ce^{x}\).
Mr.Math
  • Mr.Math
Your statement is still not correct :P
anonymous
  • anonymous
Welllll hold on
Zarkon
  • Zarkon
it is correct....find c
anonymous
  • anonymous
the e^x makes senese with f'(x) = f(x) but this function is an integral
Mr.Math
  • Mr.Math
You're saying c=0 @Zarkon.
Zarkon
  • Zarkon
\[f(0) = \int\limits_{0}^{0} f(t) dt=0\]
anonymous
  • anonymous
It wants me to prove the function is 0
Zarkon
  • Zarkon
\[\Rightarrow c=0\]
Mr.Math
  • Mr.Math
That's right! I'm a loser!! :(
anonymous
  • anonymous
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
  • Mr.Math
Exactly!
anonymous
  • anonymous
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
  • across
Mr. Michale Spivak did a good job, that is, to elicit eager students to congregate and think this one through. xd
anonymous
  • anonymous
I hate michale spivak :P
anonymous
  • anonymous
I'm just kidding its just a challening course for me
Mr.Math
  • Mr.Math
Lol @across. @Wall, this is a first order homogeneous equation and had only this solution, \(i.e f(x)=ce^{x}\).
Mr.Math
  • Mr.Math
homogeneous differential equation* and it has* *_*
anonymous
  • anonymous
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
  • Mr.Math
You're welcome! Thanks for fanning me :D

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