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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?

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 $\left|I-S_{n}\right| \leq 0.0001$

Integration Techniques

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University of Wisconsin - Madison

Integration Techniques