?The trouble with the error estimates is that it is often very difficult to compute
Chapter 7, Problem 23(choose chapter or problem)
The trouble with the error estimates is that it is often very difficult to compute fourth derivatives and obtain a good upper bound K for \(\left|f^{(4)}(x)\right|\) by hand. But mathematical software has no problem computing \(f^{(4)}\) and graphing it, so we can easily find a value for K from a machine graph. This exercise deals with approximations to the integral \(I=\int_{0}^{2 \pi} f(x) \ d x\) where \(f(x)=e^{\cos x}\). In parts (b), (d),and (g), round your answers to 10 decimal places.
(a) Use a graph to get a good upper bound for \(\left|f^{\prime \prime}(x)\right|\).
(b) Use \(M_{10}\) to approximate I.
(c) Use part (a) to estimate the error in part (b).
(d) Use a calculator or computer to approximate I.
(e) How does the actual error compare with the error estimate in part (c)?
(f) Use a graph to get a good upper bound for \(\left|f^{(4)}(x)\right|\).
(g) Use \(S_{10}\) to approximate I.
(h) Use part (f) to estimate the error in part (g).
(i) How does the actual error compare with the error estimate in part (h)?
(j) How large should n be to guarantee that the size of the error in using \(S_{n}\) is less than 0.0001?
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