SECTION 5.4 Polynomial Approximations 303
- Assume that you have a function f satisfying f(0) = 5 and for
n≥ 1 f(n)(0) =(n 2 −n1)!.
(a) Write out P 3 (x), the third-degree Maclaurin polynomial ap-
proximation off(x).
(b) Write out the Maclaurin series forf(x), including the general
term.
(c) UseP 3 (x) to approximatef(^12 ).
(d) Assuming thatf(4)(c)≤^14 for allcsatisfying 0< c <^12 , show
that
|f(^12 )−P 3 (^12 )|< 10 −^3.- The functionf has derivatives of all orders for all real numbersx.
Assumef(2) =− 3 , f′(2) = 5, f′′(2) = 3,andf′′′(2) =− 8.
(a) Write the third-degree Taylor polynomial for f aboutx= 2
and use it to approximatef(1.5).
(b) The fourth derivative off satisfies the inequality|f(4)(x)|≤ 3
for allxin the closed interval [1. 5 ,2]. Use the Lagrange error
bound on the approximation to f(1.5) found in part (a) to
explain whyf(1.5) 6 =− 5.
(c) Write the fourth-degree Taylor polynomial,P(x), forg(x) =
f(x^2 + 2) about x= 0. Use P to explain whyg must have a
relative minimum atx= 0.- Let f be a function having derivatives of all orders for all real
numbers. The third-degree Taylor polynomial forf aboutx= 2
is given by
P 3 (x) = 7−9(x−2)^2 −3(x−2)^3.(a) Findf(2) andf′′(2).
(b) Is there enough information given to determine whetherfhas
a critical point atx= 2? If not, explain why not. If so, deter-
mine whetherf(2) is a relative maximum, a relative minimum,
or neither, and justify your answer.