1. For the function f(x)V1+x, let o 0, 0.6, and2 0.9. Construct inter- or the function...
Let Xo = 0, X,= ob and X2 = 0.9 Construct interpolation polgnonial at degree at most two to approximate f(0.45) and find the absolute error if f(x) = (x+1) #
For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin π.x For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin π.x
Problem Statement: Let f(x) = V1 + x. Back in our first semester of calculus, we used a linear approximation L(a) centered at c = 0 to find an approximation to V1.2. In our second semester, we improve upon this idea by using the Taylor polynomials centered at c= 0 (or Maclaurin polynomials) for f(x) to obtain more accurate approximations for V1.2. (a) Compute Ti(x) for f(x) = V1 + x centered at c= 0. Then compute L(x) for f(x)...
Let F be a finite field with q elements. a) S -1 for every a*0 in F. how that a-1 either f* 0 or deg(f*)<q, and f* induces the same function on F as f does. function on F, then f=g. b) Let j(X)E F[X]. Show that there exists a polynomial /*(X)EF[X] such that c) Show that if two polynomials f and g, each of degree <g, induce the same Let F be a finite field with q elements. a)...
2. Let g(x) In f(x) where f(x) is a twice differentiable positive function on (0, o) such that f(x + 1) = x f(x) Then for N 1, 2, 3 find g" N+ 2 2. Let g(x) In f(x) where f(x) is a twice differentiable positive function on (0, o) such that f(x + 1) = x f(x) Then for N 1, 2, 3 find g" N+ 2
With explanation! 3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
Consider the following function. /(x)=x-5, a= 1, n= 2, 0.8SXS 1.2 (a) Approximate f by a Taylor polynomial with degree n at the number a T2(x) = (b) Use Taylor's Inequality to estimate the accuracy of the approximation x) ~ Tn(x) when x lies in the given interval. (Round your answer to six decimal places.) (c) Check your result in part (b) by graphing Rn(x) 0.6 0.4 0.2 0.6 0.4 0.2 0.9 0.9 1.2 -0.2 -0.4 -0.6 -0.2 -0.4 -0.6...
Question: Let f(x) be a function satisfying f(0) = 0, f'(0) = 5, f'(0) = -6 and |f(3)(x) = 6 for 0 5x51. Find the Taylor polynomial of degree 2 off at x = 0 and then find lim 5x-f(x) x2 x=0+ Answer: The Taylor polynomial of degree 2 off at x = 0 is P2(x) = Near x = 0, the function f(x) is equal to P2(x) plus some remainder, that is f(x) = P2(x) + R3(x).
1. Runge's function is written as f(x) = 1 25r2 (a) Develop a plot of this function for the interval from x =-1 to 1 using Matlab (no submission required). Develop the fourth-order Lagrange interpolating polynomial using equispaced function values corresponding to xi =-1,-0.5, 0, 0.5, and 1. (Note that you first need to determine the (a. ) pairs.) Use the polynomial to estimate f(0.9). (b) What is et? (c) Generate a cubic spline using the five data points from...
let f:[-pi,pi] -> R be definded by the function f(x) { -2 if -pi<x<0 2 if 0<x<pi a) find the fourier series of f and describe its convergence to f b) explain why you can integrate the fourier series of f term by term to obtain a series representation of F(x) =|2x| for x in [-pi,pi] and give the series representation DO - - - 1. Let f: [-T, 1] + R be defined by the function S-2 if-A53 <0...