Chapter 10.4, Problem 54E

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.

54. (A)    Find the fourth-degree Taylor polynomial p₄(x) 0 for f(x) = ln (l + x), and use a graphing calculator to compute the quartic regression polynomial q₄(x) for the points (0, ln 1),

$\begin{array}{l}\left(-\frac{1}{2},\ln \left(1-\frac{1}{2}\right)\right),\left(-\frac{3}{4},\ln \left(1-\frac{3}{4}\right)\right),\\ \left(-\frac{7}{8},\ln \left(1-\frac{7}{8}\right)\right),\left(-\frac{15}{16},\ln \left(1-\frac{15}{16}\right)\right).\end{array}$

(B)    Use graphical approximation techniques to find the maximum error for $-\frac{15}{16}\le x\le 0$ in approximating f(x) = ln(1 + x) by p₄(x) and by q₄(x).

(C)    Which polynomial, p₄(x) or q₄(x), gives the better approximation to

${\displaystyle {\int }_{-15/16}^0\ln \left(1+x\right)dx?}$