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1. (5 points) Define f (x) = sin(x) + cos(x) and n x2k+1[

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 ≤ 1that passes through the points (0, 0), (0, 3), and (4, 0). The functions px and py shouldbe cubic polynomials. The plot should be based on one hundred evaluations of px andpy.3. (5 points) For n = 5, 10, and 15, find the Newton interpolating polynomial pn for thefunction f(x) = 1/(1 + x2) on the interval [−5, 5]. Use equally spaced nodes. In eachcase, compute f(x) − pn(x) for 30 equally spaced points in [−5, 5] in order to see thedivergence of pn from f.4. (5 points) Let h be a “small” number. The derivative of a function f at x0 can beapproximated by a forward divided differencef′(x0) ≈ f(x0 + h) − f(x0)hand by central divided differencef′(x0)≈ f(x0 +h)−f(x0 −h).2hFor the function f(x) = sin(x), plot the error when these approximations are used toestimate f′(1) = cos(1) for h = 10−1, 10−2, . . . , 10−16. Repeat with x0 = 1 + 106π. Useloglog 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 haveany loops.

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