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