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(Solved): Free-Response Question (9pts) You may use a calculator for this question. Let f(x) be a function tha ...
Free-Response Question (9pts)
You may use a calculator for this question.
Let f(x) be a function that is differentiable for all x. The Taylor expansion for f(x) about x=0 is given
by T(x)=\sum_(k=0)^(\infty ) (-1)^(k)((k+1)x^(3k))/(k!). The first four nonzero terms of T(x) are given by T_(4)(x)=1-2x^(3)+(3x^(6))/(2!)-(4x^(9))/(3!).
aT(x) converges for all x.
bW(x) be the Taylor expansion for x^(2)f^(')(x) about x=0. Find the general term of W(x) and find
W_(3)(x).
cf(0.5)~~T_(4)(0.5) and f(1)~~T_(4)(1). Find the values of T_(4)(0.5) and T_(4)(1).
d|f(0.5)-T_(4)(0.5)| or |f(1)-T_(4)(1)| ? Give a reason for your answer.