Taylor Series Lite
The following is a brief description of Taylor’s Theorem sufficient for our purposes. It is
by no means a comprehensive tutorial. If you wish more details, you can consult any
calculus text or look up the topic on the internet.
Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712,
gives the approximation of a differentiable function near a point by a polynomial whose
coefficients depend only on the derivatives of the function at that point.
The theorem is as follows:
If is an integer and 0n≥
f
is a function which is times continuously differentiable on
the closed interval
[
n
]
,ax and times differentiable on the open interval , then 1n+
(
,ax
)
() () ()
()()
() ()
()
(
)
()
()
23
1! 2! 3! !
nnn
fa fa f a f a
f
xfa xa xa xa xa R
n
′ ′′ ′′′
=+ −+ −+ −++ −+"
Here represents the factorial of , and !n n n
R
is a remainder term which is small if
x
is
close to . There are a number of forms of the remainder term. One such form is the
Lagrange form which states that there exists a number
a
ξ
between and
a
x
such that
(
)
()
()
()
11
1!
nn
nf
Rx
n
ξ
+
a
+
=−
+.
We use the truncated Taylor series
() () ()
()
(
)()
(
)()
23
1! 2! 3!
fa fa f a
fx fa xa xa xa
′′′ ′′′
+−+−+ −
in section 1.6, problem 43.
Later in the course we will need a two-dimensional version of a truncated Taylor Series
as an approximating function. That truncated series looks like:
()()() ()
0 0 00 00 00
,,, ,
ff
f
xxyy fxy xy x xy y
xy
⎡
⎤
∂∂
⎡⎤
++ + +
⎢
⎥
⎢⎥
∂∂
⎣⎦
⎣
⎦
The right hand side of this expression is the equation for the tangent plane to the graph of
f
at
(
00
,
)
xy
. Note, for example, that
(
00
,
f
)
x
y
x
∂
∂
is the partial derivative of
f
with
respect to
x
evaluated at
()
00
,
xy
and is thus just a number.
Math 275 Skyline College
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