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Problem 23 Medium Difficulty

The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound $ K $ for $ \mid f^{(4)}(x) \mid $ by hand. But computer algebra systems have 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 = \displaystyle \int_0^{2 \pi} f(x)\ dx $,
where $ f(x) = e^{\cos x} $.
(a) Use a graph to get a good upper bound for $ \mid f^{\prime \prime}(x) \mid $.
(b) Use $ M_{10} $ to approximate $ I $.
(c) Use part (a) to estimate the error in part (b).
(d) Use the built-in numerical integration capability of your $ CAS $ 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 $ \mid f^{(4)}(x) \mid $.
(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?

Answer

a. $K=e$ or $K=2.8$
b. $M_{10} \approx 7.954926518$
c. $\left|E_{M}\right| \leq \frac{2.8(2 \pi-0)^{3}}{24 \cdot 10^{2}}=0.289391916$
d. $I \approx 7.954926521$
e. $3 \times 10^{-9}$
f. $K=4 e$ or $K=10.9$
g. $S_{10} \approx 7.953789422$
h. $\left|E_{S}\right| \leq \frac{10.9(2 \pi-0)^{5}}{180 \cdot 10^{4}} \approx 0.059299814$
i. $7.954926521-7.953789422 \approx 0.00114$
j. $n \geq 50$ to ensure that $\left|I-S_{n}\right| \leq 0.0001$

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Video Transcript

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