Chapter 11.1, Problem 88E

A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivatives at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x) = sin x on the interval [0, π].

1. a. Write the (quadratic) Taylor polynomial p₂ for f centered at $\frac{\pi }{2}$.
2. b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:

$q(0)=f(0),q\left(\frac{\pi }{2}\right)=f\left(\frac{\pi }{2}\right),\text{and}q(\pi )=f(\pi ).$

      Show that $q(x)=-\frac{4}{{\pi }^2}{x}^2+\frac{4}{\pi }x$.

1. c. Graph f, p₂, and q on [0, π].
2. d. Find the error in approximating f(x) = sin x at the points $\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}$, and π using p₂, and q.
3. e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.