3. Runges function is written as

f(x) = 1 1 + 25x 2

(a) Develop a plot of this function for the interval from x = 1 to 1 using Matlab.

(b) Generate and plot the fourth-order Lagrange interpolating polynomial using equispaced function values corresponding to xi = 1, 0.5, 0, 0.5, and 1.

(c) Use the five points from 3b to estimate f(0.8) with first- through fourth-order Newton interpolating polynomials. Use xi = 1, 1 for first order, xi = 1, 0, 1 for second order, xi = 1, 0, 0.5, 1 for third order, and xi = 1, 0.5, 0, 0.5, 1 for fourth order.

(d) Generate and plot a cubic spline using the five points from 3b. Estimate f(0.8).

(e) Discuss your results.