1. (10 pts) Let P(x) be the Lagrange Interpolating polynomial of f(x) = sin x at the points x0 = 1, x1 = 2, x2 = 3, and x3 = 4. (a) Explicitly find the Ln,j functions and express P(x) in terms of these functions. (b) Find the relative error in approximating sin(?/6) using P(?/6). (c) Show that X3 k=0 L3,k(x) = 1 (hint: Simplify) 2. (10 pts) Suppose that you want to solve x2 ! 5 = 0 on the interval [1,3]. (a) Explain why the bisection method is certain to converge. (b) Perform three iterations (that is, find p3) using Newton-Raphson method, by hand, with an initial guess of p0 = 1.5. (c) Determine the absolute error in the value p3 from (b) to the actual root is x = p5. (d) Find a bound on the number of iterations of Bisection Method that would need to be performed to match the error found in (c). (e) Use Octave to find the exact number of iterations needed to approximate the the root within a tolerance that is the error found in (c). 3. (10 pts) Let f(x) = ex. (a) Find the third degree Taylor’s polynomial, P3(x) expanded about x0 = 0. (b) Write an Octave code which will plot both functions on the interval [0,3], with one function in blue and the other in red. Print the code and output, and include these items in your exam submission. (c) Compute by hand Z 3 0 ex dx and compare to Z 3 0 P3(x) dx. 4. (10 pts) Suppose we have data points x = 0, 1, 2, 3 with corresponding y = 2, 2, 4, 5. (a) Use Divided Di?erences to determine the interpolating polynomial (by hand).