1. Show that for f(x) : 2+ ex + (e 1)sin (1%). 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 Pn(X) for f(x) about x0. Find a value n necessary for Pn(x) to approximate f(x) within 106 on [0. 0.5]. 3. Compute the actual, relative, and roundoff error of the approximations of pme by p: (a) ptrue : ?T, p : 22/7 (b) pme = W p is the floating point form of pme obtained by 3digit rounding. 4. The number e can be defined by e = 2:0(1!) where n! = n x (n 1) - - - 2- 1 for n 75 0 and O! = 1. Compute the absolute and relative error of 23:0 %. 5. Suppose two points (X0,y0) and (X1,y1) are on a straight line where yl # yo. You can use two formulas to evaluate the Xintercept of the line: * : >ll= = X0 _ o 1 yo mil/0 (a) Show that x* = X** algebraically and that X* is the Xintercept of the line con taining (x0,y0) and (X1,y1). (b) Let (X0,y0) = (1.31, 3.24) and (x1,y1) = (1.93, 4.76). Compute the xintercept using the two formulas above and threedigit rounding arithmetic. Which method is better? 6. (a) How many additions and multiplications are needed to compute the following: a = 2:23ng. [:1 j:l (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) = aan + an_1x\"1 + + alx + ao and suppose X0 is given. Construct an algorithm to evaluate P(x0) using nested multiplication. How many additions and multiplications are required