Quiz #9
Math 1220, Spring 2005
(5 points) Problem 1. (a) Find the Taylor series in x−1 through (x−1)4 of the function f (x) = ex .
(b) Find the Taylor polynomial of order 3 based at 1 for the function f (x) = ex .
(c) Find a formula for R3 (x) and obtain a good bound for |R3 (0.5)|.
f (n+1) (c)
Use that Rn (x) =
(x − a)n+1 , where c is a number between a = 1 and x.
(n + 1)!
1
2
(5 points) Problem 2.
(a) Use the Trapezoidal Rule with n = 2 to approximate
Z 1
1
dx.
2
0 1+x
(b) Determine n such that the Trapezoidal Rule will approximate the integral with an error En
1
satisfying |En | ≤
.
30000
′′
1
6x3 − 2
(b − a)3 ′′
f
(c)
with
c
between
a
and
b,
and
.
=
Use that En = −
12n2
1 + x2
(1 + x2 )3