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lgbasallote

  • 2 years ago

find f \[f''(x) = x^{-2}, \quad x > 0,\quad f(1) = 0, \quad f(2) = 0\]

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  1. myininaya
    • 2 years ago
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    To find f you need to first find f' the find f' given f'' integrate both sides

  2. lgbasallote
    • 2 years ago
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    hmm.. \[f'(x) = \frac 1x + c\] ??

  3. myininaya
    • 2 years ago
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    almost

  4. myininaya
    • 2 years ago
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    you are missing a certain constant multiple

  5. lgbasallote
    • 2 years ago
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    ahh \[f'(x) = -\frac 1x + c\]

  6. myininaya
    • 2 years ago
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    yes :) now to find the constant hmmm....you are missing a certain initial condition to do that .... your question doesn't make sense .... you need one of those to be f'(something)=another something

  7. lgbasallote
    • 2 years ago
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    or maybe i should do it \[f'(x) = -\frac 1x + c_1\]

  8. lgbasallote
    • 2 years ago
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    i might have typoed

  9. Zarkon
    • 2 years ago
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    integrate again

  10. lgbasallote
    • 2 years ago
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    ah yes i did. it's f'(2) not just f(2)

  11. Zarkon
    • 2 years ago
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    get a system of 2 eau and 2 unknowns

  12. lgbasallote
    • 2 years ago
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    hmm \[f(x) = -\ln x + c_1 x + c_2\] yes?

  13. myininaya
    • 2 years ago
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    ok great. you can find that constant by using f(1)=0 and then do what zarkon says to find f

  14. Zarkon
    • 2 years ago
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    you can do the problem with two given values of f

  15. myininaya
    • 2 years ago
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    or you can wait to find the first constant whatever

  16. lgbasallote
    • 2 years ago
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    i suppose x > 0 is just there to note that -ln x exists?

  17. myininaya
    • 2 years ago
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    oh wait.... i guess you can do it with f(something1)=another something1 and f(something2)=another something2 oops

  18. lgbasallote
    • 2 years ago
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    f'(2) = 0 so f'(2) = -1/2 + c_1 = 0 does this mean c_1 is 1/2?

  19. myininaya
    • 2 years ago
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    yes adding 1/2 to both sides solves that equation for c_1

  20. lgbasallote
    • 2 years ago
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    then f(1) = -ln (1) + 1/2 x + c_2 = 0 so c_2 is -1/2?

  21. myininaya
    • 2 years ago
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    x is 1 so you have -ln(1)+1/2(1)+c_2=0 and yes

  22. lgbasallote
    • 2 years ago
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    oh. yeah...forgot to sub 1 into x the second time

  23. lgbasallote
    • 2 years ago
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    so \[f(x) = -\ln x + \frac 12 x - \frac 12\] ??

  24. myininaya
    • 2 years ago
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    tep

  25. lgbasallote
    • 2 years ago
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    nice. thanks

  26. lgbasallote
    • 2 years ago
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    i just noticed this was my 1000th question

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