There are different ways to approximate a function f by polynomials. If, for example, f(a), f′(a), and fʺ(a) are known, then we can construct the second-degree Taylor polynomial p2(x) at a for f(x); p2(x) and f(x) will have the same value at a and the same first and second derivatives at a. If on the other hand, f(x1), f(x2), and f(x3) are known, then we can compute the quadratic regression polynomial q2(x) for the points (x1, f(x1)), (x2, f(x2)), (x3, f(x3)); q2(x) and f(x) will have the same values at x1, x2, x3. Problems 53 and 54 concern these contrasting methods of approximation by polynomials.
53. (A) Find the second-degree Taylor polynomial p2(x) at 0 for f(x) = ex, and use a graphing calculator to compute the quadratic regression polynomial q2(x) for the points (–0.1, e–01), (0. e0), and (0.1. e0.1).
(B) Use graphical approximation techniques to Find the maximum error for –0.1 ≤ x ≤ 0.1 in approximating f(x) = ex by p2(x) and by q2(x).
(C) Which polynomial, p2(x) or q2(x), gives the better approximation to
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