MAT A26
Lecture 35
1
Taylor Polynomials
Taylor polynomials are generalizations of the constant and linear approxima
tions we studied last fall. Let us see how to obtain them.
Fix
a
∈
R
. Suppose that
f
(
x
) is
n
+1 times diﬀerentiable on the interval
I
= (
a

h, a
+
h
) for some
h >
0. Let
x
∈
I
be chosen arbitrarily and then
ﬁx it (so for now we will treat
x
as a constant!) Then by FTC we may write
f
(
x
)

f
(
a
)

{z
}
=
f
0
(
x
)
the const.
approx.
=
Z
x
a
f
0
(
x
)
dx
(= the error
E
0
(
x
))
Now integrate by parts “stupidly”, taking
u
=
f
0
(
t
) and
dv
=
dt
so
v
=
t

x
. (This is ok since
x
is a constant and so
dv
=
dt
as required.) Then
f
(
x
)

f
(
a
)

{z
}
=
E
0
=
Z
x
a
f
0
(
t
)
dt
=
f
0
(
t
)(
t

x
)

x
a

Z
x
a
(
t

x
)
f
00
(
t
)
dt
=
f
0
(
x
)(
x

x
)

f
0
(
a
)(
a

x
)

Z
x
a
(
t

x
)
f
00
(
t
)
dt
= 0 +
f
0
(
a
)(
x

a
)

Z
x
a
(
t

x
)
f
00
(
t
)
dt
Transposing the ﬁrst term on the right gives
f
(
x
)

(
f
(
a
) +
f
0
(
a
)(
x

a
))

{z
}
=
f
1
(
x
)
,
the linear approx.
=

Z
x
a
(
t

x
)
f
00
(
t
)
dt
1
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View Full DocumentNow repeat this: integrating by parts “stupidly”, taking
u
=
f
00
(
t
) and
dv
= (
t

x
)
dt
so
v
=
1
2
(
t

x
)
2
. Then
f
(
x
)

(
f
(
a
) +
f
0
(
a
)(
x
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 Fall '08
 HIMONAS
 Calculus, Polynomials, Approximation, Derivative, Linear Approximation, Taylor Series, dt, Taylor's theorem

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