Consider the function f(x)=x⋅sin(x).
a) Find the area bound by y=f(x) and the x-axis over the interval, 0≤x≤π. (Do this without a calculator for practice and give the exact answer.)
b) Let M(x) be the Maclaurin polynomial that consists of the first 5 nonzero terms of the Maclaurin series for f(x). Find M(x) by taking advantage of the fact that you already know the Maclaurin series for sin x.
c) Since every Maclaurin polynomial is by definition centered at 0, the farther from zero, the less accurately it approximates its corresponding function. If you were to approximate the area you calculated in part (a) by correctly integrating M(x) found in part (b) the resulting estimate would be off by more than 0.0001 from the actual area. (That is, ∣∣∣∣∫0πf(x)dx−∫0πM(x)dx∣∣∣∣>0.0001).
Find the largest a (to three decimal places) between x=0 and x=π
so that the alternating series error bound guarantees
(Give a number with exactly three digits after the decimal point.)
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