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25 September, 07:06

Use a transformation to solve the equation. f ∙ 8 = 40 a f=5 b f=32 c f=48 df=320

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  1. 25 September, 07:18
    0
    Using Taylor expansion, show that

    f0

    (x0) = f (x0 + h) - f (x0)

    h - h

    2

    f00 (ξ),

    for some ξ lying in between x0 and x0 + h.

    Solution: We expand the function f in a first order Taylor polynomial around x0:

    f (x) = f (x0) + (x - x0) f0

    (x0) + (x - x0)

    2 f00 (ξ)

    2,

    where ξ is between x and x0. Let x = x0 + h:

    f (x0 + h) = f (x0) + hf0

    (x0) + h2

    2 f00 (ξ).

    Solving for f0

    (x0), we obtain:

    f0

    (x0) = f (x0 + h) - f (x0)

    h - h

    2

    f00 (ξ)
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