## Walleye 3 years ago Prove that if f(x) = integral from 0 to x of f(t) dt then f = 0

1. Walleye

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

2. Mr.Math

This is not a true statement.

3. Walleye

Michael Spivak claims it is

4. Mr.Math

Who's Michael Spivak?

5. Walleye

The man who wrote my textbook.

6. Mr.Math

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

7. Mr.Math

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

8. imranmeah91

$\int_0^x 2t dt= t^2$ $x^2$

9. Mr.Math

Oh wait!

10. Walleye

haha im not going anywhere with this one

11. across

It implies f(0)=0.

12. Walleye

hes claiming f(x) = 0 for any x

13. Walleye

I certinally dont see it

14. imranmeah91

15. Walleye

Nope that is the whole question

16. 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.

17. Walleye

yes f'(x) = f(x)

18. across

We know that$f(x)=\int f(x)dx\implies f(x)=e^x$^^

19. Mr.Math

Yeah.

20. Walleye

oh nice

21. Walleye

I didnt think about e^x with this one

22. Mr.Math

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

23. Mr.Math

Your statement is still not correct :P

24. Walleye

Welllll hold on

25. Zarkon

it is correct....find c

26. Walleye

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

27. Mr.Math

You're saying c=0 @Zarkon.

28. Zarkon

$f(0) = \int\limits_{0}^{0} f(t) dt=0$

29. Walleye

It wants me to prove the function is 0

30. Zarkon

$\Rightarrow c=0$

31. Mr.Math

That's right! I'm a loser!! :(

32. Walleye

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

33. Mr.Math

Exactly!

34. Walleye

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)

35. across

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

36. Walleye

I hate michale spivak :P

37. Walleye

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

38. Mr.Math

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

39. Mr.Math

homogeneous differential equation* and it has* *_*

40. Walleye

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!!!!

41. Mr.Math

You're welcome! Thanks for fanning me :D