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Walleye

  • 3 years ago

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

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  1. Walleye
    • 3 years ago
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    if \[f(x) = \int\limits_{0}^{x} f(t) dt \] then f = 0

  2. Mr.Math
    • 3 years ago
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    This is not a true statement.

  3. Walleye
    • 3 years ago
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    Michael Spivak claims it is

  4. Mr.Math
    • 3 years ago
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    Who's Michael Spivak?

  5. Walleye
    • 3 years ago
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    The man who wrote my textbook.

  6. Mr.Math
    • 3 years ago
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    Whoever he might be, tell him Newton has another opinion :P

  7. Mr.Math
    • 3 years ago
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    This is a direct use of the fundamental theorem of calculus.

  8. imranmeah91
    • 3 years ago
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    \[\int_0^x 2t dt= t^2 \] \[x^2\]

  9. Mr.Math
    • 3 years ago
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    Oh wait!

  10. Walleye
    • 3 years ago
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    haha im not going anywhere with this one

  11. across
    • 3 years ago
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    It implies f(0)=0.

  12. Walleye
    • 3 years ago
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    hes claiming f(x) = 0 for any x

  13. Walleye
    • 3 years ago
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    I certinally dont see it

  14. imranmeah91
    • 3 years ago
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    is there derivative sign infront of integral?

  15. Walleye
    • 3 years ago
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    Nope that is the whole question

  16. Mr.Math
    • 3 years ago
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    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
    • 3 years ago
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    yes f'(x) = f(x)

  18. across
    • 3 years ago
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    We know that\[f(x)=\int f(x)dx\implies f(x)=e^x\]^^

  19. Mr.Math
    • 3 years ago
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    Yeah.

  20. Walleye
    • 3 years ago
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    oh nice

  21. Walleye
    • 3 years ago
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    I didnt think about e^x with this one

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

  23. Mr.Math
    • 3 years ago
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    Your statement is still not correct :P

  24. Walleye
    • 3 years ago
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    Welllll hold on

  25. Zarkon
    • 3 years ago
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    it is correct....find c

  26. Walleye
    • 3 years ago
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    the e^x makes senese with f'(x) = f(x) but this function is an integral

  27. Mr.Math
    • 3 years ago
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    You're saying c=0 @Zarkon.

  28. Zarkon
    • 3 years ago
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    \[f(0) = \int\limits_{0}^{0} f(t) dt=0\]

  29. Walleye
    • 3 years ago
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    It wants me to prove the function is 0

  30. Zarkon
    • 3 years ago
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    \[\Rightarrow c=0\]

  31. Mr.Math
    • 3 years ago
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    That's right! I'm a loser!! :(

  32. Walleye
    • 3 years ago
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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

  33. Mr.Math
    • 3 years ago
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    Exactly!

  34. Walleye
    • 3 years ago
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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)

  35. across
    • 3 years ago
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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

  36. Walleye
    • 3 years ago
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    I hate michale spivak :P

  37. Walleye
    • 3 years ago
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    I'm just kidding its just a challening course for me

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

  39. Mr.Math
    • 3 years ago
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    homogeneous differential equation* and it has* *_*

  40. Walleye
    • 3 years ago
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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!!!!

  41. Mr.Math
    • 3 years ago
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    You're welcome! Thanks for fanning me :D

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