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
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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