1. (5 points) Define f (x) = sin(x) + cos(x) and n x2k+1
[0, 2π].
2. (5 points) Write a Matlab script that plots a closed curve
(px(t), py(t)), 0 ≤ t ≤ 1
that passes through the points (0, 0), (0, 3), and (4, 0). The functions px and py should be cubic polynomials. The plot should be based on one hundred evaluations of px and py.
3. (5 points) 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 nodes. In each case, compute f(x) − pn(x) for 30 equally spaced points in [−5, 5] in order to see the divergence of pn from f.
4. (5 points) Let h be a “small” number. The derivative of a function f at x0 can be approximated by a forward divided difference
f′(x0) ≈ f(x0 + h) − f(x0) h
and by central divided difference
f′(x0)≈ f(x0 +h)−f(x0 −h).
2h
For the function f(x) = sin(x), plot the error when these approximations are used to estimate f′(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.