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

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Anna Marie V.

Campbell University

Kristen K.

University of Michigan - Ann Arbor

Boston College

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Topics

Integration Techniques

Anna Marie V.

Campbell University

Kristen K.

University of Michigan - Ann Arbor

Boston College

Lectures

Join Bootcamp