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Which expression yields the best approximation to df/dx (h 1)?

  1. Nov 29, 2007 #1
    Some interesting calculus...

    Which of the following expressions yields the best approximation to
    df/dx (h<<1)?

    A. [tex]\frac{f(x+h)-f(x)}{h}[/tex]

    B. [tex]\frac{f(x+\frac{h}{2})-f(x-\frac{h}{2})}{h}[/tex]

    C. [tex]\frac{f(x)-f(x-h)}{h}[/tex]

    D. [tex]\frac{f(x+h)-f(x-h)}{h}[/tex]

    From school days I have been taught A which is almost the same as C

    I would like B to be the correct.What do other people think?
  2. jcsd
  3. Nov 29, 2007 #2


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    You can use simple substitution to show that A, B and C are equivalent.
  4. Nov 29, 2007 #3
    You are correct:

    I will go for A now...

    It looks that all A,B and C are equivalent.But definition of y' is to find the increment in y at x=x.Say,y changes to y+dy and x changes to x+dx.

    My point is that to find df/dx at x,we must take increment f(x+h) from f(x)

    Remember,by definition, df/dx is evaluated at x.
  5. Nov 29, 2007 #4


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    When going to the limit A,B,C give the same result (as long as the derivative is continuous at x). However if you want an approximation where h is fixed and small then B is the best approximation.

    D is just wrong - you need to divide by 2 (then identical to B).
  6. Nov 29, 2007 #5
    Are we supposed to assume that the same h is used in each expression? In that case there would be a unique right answer to the original question, but you would need to know something about f to say which it is...
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