Numerical and Graphical Approximations (a) Use the Maclaurin polynomials P 1 ( x ) , P 3 ( x ) , and P 5 ( x ) for f ( x ) = sin x to complete the table. X 0 0.25 0.50 0.75 1 sin x 0 0.2474 0.4794 0.6816 0.8415 P 1 ( x ) P 3 ( x ) P 5 ( x ) (b) Use a graphing utility to graph f ( x ) = sin x and the Maclaurin polynomials in part (a). (c) Describe the change in accuracy of a polynomial approximation as the distance from the point where the polynomial is centered increases.
Solution Summary: The author explains that the Maclaurin polynomial for f is p_n(x)=f' (0 )x+
Comparing maclaurin Polynomials (a) Compare the Maclaurin polynomials of degree 4 and degree 5, respectively, for the functions f(x) = ex and g(x) = xex . What is the relationship between them? (b) Use the result in part (a) and the Maclaurin polynomial of degree 5 for f(x) = sin x to find a Maclaurin polynomial of degree 6 for the function g(x) = x sin x. (c) Use the result in part (a) and the Maclaurin polynomial of degree 5 for f(x) = sin x to find a Maclaurin polynomial of degree 4 for the function g(x) = (sin x)/x
2) for function f (x) = ln x
(Table attached in to description)
Table is given.
a) Construct the Newton Divided difference table for the function f(x)= lnx.
b) Write the polynomial P3(x) created by the Newtonian divided difference.
c) Calculate the exact value at the point x = 2.5 and its approximate value separately.
numerical solution
B. Find the interpolating polynomial of the following point and function using Lagrange interpolation process.1. (-3,0), (-1,2), (0,-2), (1,3) and (3,-1).
2. y = cos ( 2x ) at x = [0,π] with four equally spaced points.
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