Use graphs to find a Taylor polynomial Pn(x) for In (1 + x) so that |Pn(x) - ln (1 + x)|< 0.001 for every x in [ - 0.5,0.5]. Use graphs to find a Taylor polynomial Pn(x) for cos x so that |Pn(x) - cos x|< 0.001 for every x in [ - pi, pi]. Find a formula for the truncation error if we use P6(x) to approximate 1/1 - 2x on ( - 1/2, 1/2). Find a formula for the truncation error if we use P9(x) to approximate 1/1 - x on ( - 1, 1) In Exercises 15 - 18, use the Remainder Estimation Theorem to prove that the Maclaurin series converges to the generating function from the given exercise. Exercise 7 Exercise 6 Exercise 9 Exercise 8 For approximately what values of x can you replace sin x by x - (x3/6) with an error magnitude no greater than 5 times 10 - 4? Give reasons for your answer. In Exercises 27 - 31, find the linearization and the quadratic approximation of f at x = 0. Then graph the function and its linear and quadratic approximations together around x = 0 and comment on how the graphs are related. f(x) = In (cos x) f(x) = 1/ 1 - x2 f(x) = tan x A Cubic Approximation of ex The approximation ex 1 + x + x2/2 + x3/6 is used on small intervals about the origin Estimate the magnitude of the approximation error for |x| 0.1. A Cubic Approximation Use the Taylor polynomial of order 3 to find the cubic approximation of f(x) = 1/(1 - x) at x = 0. Give an upper bound for the magnitude of the approximation error for |x| 0.1.
