Estimating error Refer to Theorem 7.2 and let f(x) = sin

Chapter 5, Problem 55AE

(choose chapter or problem)

Estimating error  Refer to Theorem 7.2 and let \(f(x)=\sin e^{x}\).

(a) Find a Trapezoid Rule approximation to \(\int_{0}^{1} \sin \left(e^{x}\right) d x\) using n=40 subintervals.

(b) Calculate \(f^{\prime \prime}(x)\).

(c) Explain why \(\left|f^{\prime \prime}(x)\right|<6\) on [0,1], given that e<3. (Hint: Graph \(f^{\prime \prime}\).)

(d) Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 7.2.

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