Proof in Numerical Analysis: Taylor's Theorem
Solve the following problem:
Suppose that y(x)' : f(x,y(x)) on the interval [x0, x1] with y (x0) = y0. Assume that a unique solution y exists such that it and all of its derivatives up to and including the third order are defined and continuous on [x0, x1]. Using Taylor's Theorem (and the Mean Value Theorem, if necessary) prove that...
y(x1)=y(x0)+h/2[f(x0,y(x0))+f(x1,x(x1))]+ε
Where h=x1-x0 and the error term ε satisfies
|ε|≤ch3,
Where c>0 is a constant that is independent of h.(This is the mathematical way of saying ε=O(h3).)
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