MAT 581           MATLAB  TEST  1 

 Spring 2019 MAT 581代写

NAME:  

MAT 581代写 For n = 5, 10, and 15, find the Newton interpolating polynomial pn for the function f (x) = 1/(1 + x2) on the interval [−5, 5]. Use equally spaced

1.(5points) Define f (x) = sin(x) + cos(x) and MAT 581代写

Write a Matlab script that plots the functions f (x), S₁₀(x), and S₂₀(x) across the interval [0, 2π].

2. (5points) Write a Matlab script that plots a closed curve

(p_x(t), p_y(t)), 0 ≤ t ≤ 1  MAT 581代写

that passes through the points (0, 0), (0, 3), and (4, 0).  The functions p_x  and p_y  should  be cubic polynomials. The plot should be based on one hundred evaluations of  p_x  and  p_y.

3.(5 points) For n = 5, 10, and 15, find the Newton interpolating polynomial p_nfor the function f (x) = 1/(1 + x²) on the interval [−5, 5]. MAT 581代写

Use equally spaced nodes. In each case, compute f (x) − p_n(x) for 30 equally spaced points in [−5, 5] in order to see the divergence of p_n from f .

4.(5 points) Let h be a “small” The derivative of a function f at x₀can be approximated by a forward divided difference

and by central divided difference

For the function f (x) = sin(x), plot the error when these approximations are used to estimate f ^j(1) = cos(1) for h = 10⁻¹, 10⁻², . . . , 10⁻¹⁶. Repeat with x₀ = 1 + 10⁶π. Use loglog for each of the four plots and display them all in the same window using subplot. Be sure to title each plot and label the axes appropriately. Your script should not have any loops.