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    Fundamental differential equation analysis

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    If I be an open interval containing the point x. (x0) and suppose that the function f:I->R has two derivatives. Prove that

    lim as h->0 (f(x.+h) - 2f(x.) + f(x.-h))/ h^2 = f"(x.)

    © BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/fundamental-differential-equation-analysis-9565

    SOLUTION This solution is FREE courtesy of BrainMass!

    By forward difference
    f"(x.)= (f'(x+h)-f'(x))/h -----------(1)

    By backward difference method
    f'(x+h)=( f(x+h)-f(x))/h
    and f'(x)= (f(x)-f(x-h))/h

    Substituting these in equation (1)

    f"(x.)=(((f(x+h)-f(x))/h)-((f(x)-f(x-h))/h) /h

    =( f(x+h)- 2 f(x) + f(x-h) )/ h^2

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf>
    https://brainmass.com/math/calculus-and-analysis/fundamental-differential-equation-analysis-9565

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