Unformatted text preview: val [−π, π ]. Note again the scale for the y axis. It is clear that near 0, T13,0 (x) and sin(x) are essentially
indistinguishible. In fact, we will soon be able to show that for x ∈ [−1, 1],
 sin(x) − T13,0 < 10−12
while for x ∈ [−0.01, 0.01],
 sin(x) − T13,0 < 10−42 .
Indeed, in using T13,0 (x) to estimate sin(x) for very small values of x, roundoﬀ errors and
the limitations of the accuracy in ﬂoatingpoint arithmetic become much more signiﬁcant
than the true diﬀerence between the functions.
Example.
The function f (x) = ex is particularly well suited to the process of creating estimates using
Taylor polynomials. This is because for any k , the k th derivative of ex is again ex . This
means that for any n,
n Tn,0 (x) =
k=0
n =
k=0
n =
k=0 DE Math 128 f k (a)
(x − a)k
k!
e0
(x − 0)k
k!
xk
.
k! 323 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series In particular,
T0,0 (x) = 1, T1,0 (x) = 1 + x, T2,0 (x) = 1+x+ T3,0 (x) = T4,0 (x) = x2
,
2
x2
1+x+
+
2
x2
1+x+
+
2 x...
View
Full Document
 Spring '09
 Math, Derivative, Power Series, Taylor Series, Sequences And Series, Taylor's theorem, B. Forrest

Click to edit the document details