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Math Help - Linearization of f'(a).

  1. #1
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    Linearization of f'(a).

    (a) Prove if f'(a) exists, then f(x) = L(x) + e(x), where L(x) = f(a) + f'(a)(x - a) and \displaystyle\lim_{x\to a}\frac{e(x)}{x - a} = 0.

    (b) L(x) is called the linearization of f(x) at a; explain part (a) geometrically by interpreting f(x), L(x) and e(x) graphically.
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  2. #2
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    a) The usual proof for this (at least the one from Spivak's Calculus) uses L'Hopital's rule, but going through it I realized I'm not sure if it's valid if we only assume f'(a) exists (and not that the derivative exists in some neighbourhood of a).

    b)Just notice that L is the straight line tangent to f at a.
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