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A18 (estimation of the remainder for exp) Let f: R R be given through f (x) = exp(x). a) Find the Taylor polynomial Tn,xx (h) and the related remainder Rn,x* (h) of f for the expansion point Xx = 0. b) Give a bound for the remainder R20,0(h) I for X E [-5,5]. A19 (Taylor polynomial of the inverse function) Let f: (1,00) 1,00), f (x) = X In X - X. Show that f is bijective and twice differentiable. Compute the first two derivatives of the inverse function f -1 - at the point y = 0 and calculate the second order Taylor polynomial of f -1 for the expansion point Y* = A20 (Newtons Method) Use the Newton method to determine an approximate zero for the function g(x) = (x - 1) (cos X + 2) starting from the starting values 2.5 and 105. . What is the error in the sixth iteration step I (x6 - 1|?

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