A different kind of approximation When approximating a

Chapter 9, Problem 98

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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 1x2 = sin x on the interval 30, p4. a. Write the (quadratic) Taylor polynomial p2 for f centered at p 2 b. Now consider a quadratic interpolating polynomial q1x2 = ax2 + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied: q102 = f 102, qap 2 b = f ap 2 b, and q1p2 = f 1p2. Show that q1x2 = - 4 p2 x2 + 4 p x. c. Graph f, p2, and q on 30, p4. d. Find the error in approximating f 1x2 = sin x at the points p 4 , p 2 , 3p 4 , and p using p2 and q. e. Which function, p2 or q, is a better approximation to f on 30, p4? Explain.