14 91 Approximate 79 using a Taylor polynomial with n 3 Comp

14. (9.1) Approximate 79 using a Taylor polynomial with n 3. Compute the absolute error in the approximation a the exact value is given by a calculator.

Solution

let f(x) = x^1/4

taylor polynomial

T(x) = f(a) + f\'(a) * ( x-a) + 1/2 f\"(a) * (x-a)^2 + 1/3 f\'\'\'(a) * (x-a)^3

f\'(x) = 1/4 x^ -3/4 , f\'(81) = -1/108

f\"(x) = -3/16 x^ -7/4 = -1/11664

f\'\'\'(x) = 21/64 x^-11/4 = -21 / 11337408

a = 81

plugging the values in the formula

T(x) = f(a) + f\'(a) * ( x-a) + 1/2 f\"(a) * (x-a)^2 + 1/3 f\'\'\'(a) * (x-a)^3

T(x) = 3 + ( -1/108) ( 79-81) + 1/2 ( -1/11664) (79-81)^2 + 1/3 (-21 / 11337408 ) ( 79-81)^3

T(x) = 3 + .0185 -.000171 + .000004939

T(x) = 2.98