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Walleye Group Title

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

  • 2 years ago
  • 2 years ago

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

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

    • 2 years ago
  3. Walleye Group Title
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    Michael Spivak claims it is

    • 2 years ago
  4. Mr.Math Group Title
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    Who's Michael Spivak?

    • 2 years ago
  5. Walleye Group Title
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    The man who wrote my textbook.

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

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

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

    • 2 years ago
  9. Mr.Math Group Title
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    Oh wait!

    • 2 years ago
  10. Walleye Group Title
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    haha im not going anywhere with this one

    • 2 years ago
  11. across Group Title
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    It implies f(0)=0.

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

    • 2 years ago
  13. Walleye Group Title
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    I certinally dont see it

    • 2 years ago
  14. imranmeah91 Group Title
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    is there derivative sign infront of integral?

    • 2 years ago
  15. Walleye Group Title
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    Nope that is the whole question

    • 2 years ago
  16. Mr.Math Group Title
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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.

    • 2 years ago
  17. Walleye Group Title
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    yes f'(x) = f(x)

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

    • 2 years ago
  19. Mr.Math Group Title
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    Yeah.

    • 2 years ago
  20. Walleye Group Title
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    oh nice

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

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

    • 2 years ago
  23. Mr.Math Group Title
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    Your statement is still not correct :P

    • 2 years ago
  24. Walleye Group Title
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    Welllll hold on

    • 2 years ago
  25. Zarkon Group Title
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    it is correct....find c

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

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

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

    • 2 years ago
  29. Walleye Group Title
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    It wants me to prove the function is 0

    • 2 years ago
  30. Zarkon Group Title
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    \[\Rightarrow c=0\]

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

    • 2 years ago
  32. Walleye Group Title
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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

    • 2 years ago
  33. Mr.Math Group Title
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    Exactly!

    • 2 years ago
  34. Walleye Group Title
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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)

    • 2 years ago
  35. across Group Title
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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

    • 2 years ago
  36. Walleye Group Title
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    I hate michale spivak :P

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

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

    • 2 years ago
  39. Mr.Math Group Title
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    homogeneous differential equation* and it has* *_*

    • 2 years ago
  40. Walleye Group Title
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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!!!!

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

    • 2 years ago
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