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Taylor Approximation

  1. Mar 29, 2014 #1

    Nugso

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    1. The problem statement, all variables and given/known data

    Show that [tex]∫f'(x)dx/f(x) = ln|(f(x)|+C[/tex] where f(x) is a differential function.


    2. Relevant equations

    First order Taylor approximation? [tex]f(x)=f(a)+f'(a)(x-a)[/tex]


    3. The attempt at a solution

    Well, I'm not really sure how to approach the question. It's my Numerical Methods homework, so I think I have to do it by using Taylor approximation. By applying the first order Taylor approximation I get:

    [tex]ln|f(x)|=y, ln|f(a)| - ln|(f(x)| = (a-x)f'(x)/f(x)[/tex]

    [tex]ln|(f(a)/f(x)| = (a-x)f'(x)/f(x)[/tex]

    I'm kind of stuck here. Am I right in thinking that Taylor approximation is an appropriate way to approach the question?
     
  2. jcsd
  3. Mar 29, 2014 #2

    Ray Vickson

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    Start again: you are on the wrong track.
     
  4. Mar 29, 2014 #3

    Nugso

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    Hi Ray, thanks for the reply. Would you mind giving me a hint? I really haven't a clue about the right track.
     
  5. Mar 29, 2014 #4

    micromass

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    Why not take the derivative of the right-hand side?
     
  6. Mar 29, 2014 #5

    Nugso

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    I guess my teacher wants us to solve it by the difficult way, i.e using appropriate ways.
     
  7. Mar 29, 2014 #6

    Ray Vickson

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    There is nothing wrong with using the Fundamental Theorem of Calculus to solve the problem. To verify that ##\int f(x) \, dx = F(x) + C##, just check that ##F'(x) = f(x)##.

    That is an absolutely 100% correct way to do the question. In fact, such checking should always be done out of habit, whenever you are faced with a possible formula for the indefinite integral F(x) of an integrand f(x). That is a good way to catch errors.
     
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