MATH 106D - CALCULUS II
WINTER 2005
QUIZ 6*
NAME:
Show ALL your work CAREFULLY.
Consider the function
f(x)= 1
1+x.
(a) Find the degree four Taylor polynomial P4(x) for approximating f(x)
near x=0.
Write f(x)=(1+x)โ1. Then, f๎(x)=โ(1 + x)โ2,f๎๎(x)=2(1+
x)โ3,f๎๎๎(x)=โ3!(1 + x)โ4and f(4)(x)=4!(1+x)โ5. Evaluating these
derivatives at x=0,weobtainf๎(0) = โ1,f๎๎(0) = 2,f๎๎๎(0) = โ3! and
f(4)(0) = 4!.Notethatf(0) = 1. It follows that
P4(x)=f(0) + f๎(0)x+f๎๎(0)
2x2+f๎๎๎(0)
3! x3+f(4)(0)
4! x4
=1โx+x2โx3+x4.
(b) Find the degree three Taylor polynomial P3(x; 1) for approximating
f(x)nearx=1.
Using the derivatives from (a), f๎(1) = โ1
4,f๎๎(1) = 1
4,f๎๎๎(1) = โ3!
16
and f(1) = 1
2. It follows that
P3(x;1) = f(1) + f๎(1)(xโ1) + f๎๎(1)
2(xโ1)2+f๎๎๎(1)
3! (xโ1)3
=1
2
โ
(xโ1)
4+(xโ1)2
8
โ
(xโ1)3
16 .
(c) Write down the Taylor series (using summation sign, i.e., you need to
find the general n-th term of the series) for f(x)nearx=0.
From part (a), we see that the general n-th term is given by
(โ1)nxnso that the desired Taylor series is given by
โ
๎
n=0
(โ1)nxn.
Date: March 7, 2005.
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