1. Show that for f(x) = 2 + e* + (e 1)sin (TX). - 2 f'(x) is 0 at least once in the interval [0, 1]. 2. Let f(x) = (1-x)-1 and x0 = 0. Find the nth Taylor Polynomial P(x) for f(x) about xo. Find a value n necessary for P(x) to approximate f(x) within 10-6 on [0, 0.5]. 3. Compute the actual, relative, and round-off error of the approximations of Ptrue by p: (a) Ptrue = , p = 22/7 (b) Ptrue = 2, p is the floating point form of Ptrue obtained by 3-digit rounding. 4. The number e can be defined by e = (1/n!) where n! = n (n 1)...2.1 for n0 and 0! = 1. Compute the absolute and relative error of =0 5 1 5. Suppose two points (X, Yo) and (1, 1) are on a straight line where y yo. You can use two formulas to evaluate the x-intercept of the line: x* = ** 1-10 1- = Xo (xo-x1)yo 1- (a) Show that x* = x** algebraically and that x* is the x-intercept of the line con- taining (xo, yo) and (x1,y1). (b) Let (xo, yo) = (1.31, 3.24) and (X1, 1) = (1.93, 4.76). Compute the x-intercept using the two formulas above and three-digit rounding arithmetic. Which method is better? 6. (a) How many additions and multiplications are needed to compute the following: a = m n - aibi i=1 j=1 (b) Modify the formula for a so that you get the same value but reduce the number of computations. How many additions and multiplications are needed now? 7. Let P(x) = anx" + an1xn1 + ... + ax+ao and suppose X is given. Construct an algorithm to evaluate P(x) using nested multiplication. How many additions and multiplications are required?