Chapter 10.3, Problem 11E

(2) Construct the Simpson 3/8-th formula to approximate / f(x)dr by using La- grange interpolating polynomial.

Find the Taylor polynomial of degree 3 for the function f(x) = X T3(x) = + (x − 3) + (x − 3)² + (x − 3)³ about x = 3.

Use the second Taylor polynomial P2 (x) for f(x) = x? + In(1 + x) about x, = 0 and evaluate f f(x)dr. O 1

Fill in the coefficients of the degree 3 Taylor polynomial for f(x) = sin x centered at x = 3 T3(x) = [ Select ] %D [ Select ] - (x – ) + [ Select ] (x - 5)² 3 [ Select ] - (x - 5)* IT 3

Use the third Taylor polynomial P3(x) for f(x) = x + xlnx about x0 = 1 and evaluate f(2).

One of the two sets of functions, f1, f2, f3, or g1, 92, 93, is graphed in the figure to the left below; the other set is graphed in the right figure. Points A and B each have x = 0. Taylor polynomials of degree 2 approximating these functions near x = 0 are as follows: f1(x) - 5 +x + 2x? 91(x) - 2 - x + 2x2 f2(x) 2 5 + x – x2 92(x) z 2 + x + 2x? f3(x) 2 5 + x + x? 93(x) z 2 – x + x². II. B (a) Match the functions to the appropriate figure: f matches || g matches | v (b) What are the coordinates of the points A and B? A = В - (c) Match each function with the graphs (a)--(c) in the appropriate figure. fi = ? f2 ? f3 = ? 91 = ? 92 = ? 93 = ? > > > ||

Let P(x) = 3– 3x² +6x* be the fourth-degree Taylor polynomial for the function about x = 0. What is the value of " (0),

Given the following set S of polynomials, find a polynomial p(x) in P4 so that SU {p(x)} spans P4 and a non-zero polynomial q(x) in P4 so that SU{q(x)} does not span P4. Use the '^' character to indicate an exponent, e.g. 5x^2-2x+1. S = 3 ·|-4x4-2x³−x²+4x−6, −6x²+4x³+3x²+7x−9, 2x²−7x³−x²-x+10, −4x4+10x³+3x²−6x| p(x) = 0 _q(x) = 0

2.33 Let p(r) = x4 + x³ -- 2x2 + x + 1. Prove that the polynomial equation p(x) = 0 has a root in (-1,0).