Find the best approximation to the first derivative of f ( x ) based on the values of f ( x ) , f ( x + h ) , f ( x + 2 h ). What is the accuracy of this approximation?

Question

Find the best approximation to the first derivative of   f  (  x  ) based on the values of   f  (  x  )  ,  f  (  x  +  h  )  ,  f  (  x  +  2  h  ). What is the accuracy of this approximation?

Kaydence Washington · Accepted Answer

f    ′    (  x  )  ≈                    f        (        x        +        h        )        −        f        (        x        )            h        for small h but that may be closer to       f    ′    (  x  +      h    2    ).So if you have a reasonable approximation to both       f    ′    (  x  +            3      h        2    ) and       f    ′    (  x  +      h    2    ) and if you think you function is fairly smooth (in the sense of an almost constant second derivative in the range   [  x  ,  x  +  2  h  ] then you might reduce the earlier estimate by half the difference between those two, giving      f    ′    (  x  )  ≈      f    ′    (  x  +            h      2        )  −      1    2        (          f      ′        (    x    +                            3          h                2              )    −          f      ′        (    x    +                  h        2              )    )  and            f      (      x      +      h      )      −      f      (      x      )        h    −      1    2        (                            f          (          x          +          2          h          )          −          f          (          x          +          h          )                h              −                            f          (          x          +          h          )          −          f          (          x          )                h              )    =            4      f      (      x      +      h      )      −      3      f      (      x      )      −      f      (      x      +      2      h      )              2      h        .In fact this is the only estimate which gives correct results for quadratic functions and any   h.

minwaardekn · Accepted Answer

From the Taylor series,   f  (  x  +  h  )  =  f  (  x  )  +  h      f    ′    (  x  )  +      h    2        f    ″    (  x  )      /    2  +  O  (      h    3    ) and   f  (  x  −  h  )  =  f  (  x  )  −  h      f    ′    (  x  )  +      h    2        f    ″    (  x  )      /    2  +  O  (      h    3    )The only combination which gets rid of the       f    ″    (  x  ) term is             f      (      x      +      h      )      −      f      (      x      −      h      )              2      h        =      f    ′    (  x  )  +  O  (      h    2    )

Find the best approximation to the first derivative of f ( x ) based on the values of

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Answer & Explanation

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