we used Taylor series expansions of f(x0 + h) and f(x0 − h) along with Taylor’s theorem to show that f 0 (x0) = f(x0 + h) − f(x0 − h) 2h + O(h^2 ). Assuming f satisfies the conditions of Taylor’s theorem on a sufficiently large interval around x0, show that f'''(x0) = −f(x0 − 2h) + 2f(x0 − h) − 2f(x0 + h) + f(x0 + 2h)/2h^3 + O(h^2 ).
we used Taylor series expansions of f(x0 + h) and f(x0 − h) along with Taylor’s theorem to show that f 0 (x0) = f(x0 + h) − f(x0 − h) 2h + O(h^2 ). Assuming f satisfies the conditions of Taylor’s theorem on a sufficiently large interval around x0, show that f'''(x0) = −f(x0 − 2h) + 2f(x0 − h) − 2f(x0 + h) + f(x0 + 2h)/2h^3 + O(h^2 ).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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we used Taylor series expansions of f(x0 + h) and f(x0 − h) along with Taylor’s theorem to show that f 0 (x0) = f(x0 + h) − f(x0 − h) 2h + O(h^2 ). Assuming f satisfies the conditions of Taylor’s theorem on a sufficiently large interval around x0, show that f'''(x0) = −f(x0 − 2h) + 2f(x0 − h) − 2f(x0 + h) + f(x0 + 2h)/2h^3 + O(h^2 ). Your proof should include appropriate Taylor series expansions of f(x0−2h), f(x0−h), f(x0+h) and f(x0+2h) as well as a clear indication of when you use Taylor’s theorem
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