1, Verify that the zero of f(x) = 0 is the fixed point of the iteration function in Newton's method. 2, Solve the equation f(x) = e"{2x} + x - 4 = 0 by Newton's method on the interval [0.5, 1] by the two choices of x0 = 0.5 and 1, Check if they satisfy the condition of Theorem 5 first. Report your number of iterations and the zero up to 0.00001. 3, Using Newton's method to find the zero of f(x) = x/'3 - 2x - 5 = 0, the equation that Sinlsaac Newton studied in 1669. Start by locating the zero on a small enough interval. Report your number of iterations and the zero up to 0.00001. 4, (a) Verify the formula for the determinant of the 3 by 3 Vandermonte matrix [1, x, x"2; 1, y, y/\\2; 1, z, 2A2] as (x-y)(x-z)(y-z). (b) [Optional] Verify the n by n case. 5, Use minomial basis functions to interpolate the data (1 , 0), (-1, -3), and (2, 4) by a polynomial of degree no more than 2. 6, Use Lagrange basis functions to interpolate the following data: (0, 2), (1 , 3), (2, 0), and (3, -1) by a polynomial of degree of no more than 3. 7, Predict the value of e"{-0.5} by considering interpolating the function y=e'\\{-x} at x=0 and x=1 with a polynomial of degree one. Use your calculator to find the remainder/error of you prediction. 8, Construct the Newton's divided difference table for Problem 5 and Problem 6; write down the corresponding interpolation polynomials, simplify, and verify that they are identical to your answers earlier