Calculus
Stewart
Dr. Berg
Summer ‘09
Page 1
12.10
12.10
Taylor Series
Taylor Polynomials
Consider the problem of using a polynomial to approximate a function. Suppose
we wish to use a third order polynomial to calculate values for the exponential function
f
(
x
)
=
e
x
. We cannot expect the polynomial to be accurate over the entire domain, so we
make it as accurate as we can near some point, say zero.
First: Let
P
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
be the third order polynomial. Since we want
accuracy near zero and
f
(0)
=
e
0
=
1
, we need
P
(0)
=
a
0
+
a
1
0
+
a
2
0
2
+
a
3
0
3
=
a
0
=
e
0
=
1
so
a
0
=
1
.
Next:
We want the polynomial to have the same slope at zero as
f
(
x
)
=
e
x
and
′
f
(0)
=
e
0
=
1
, so we need
′
P
(0)
=
a
1
+
2
a
2
x
+
3
a
3
x
2
x
=
0
=
a
1
=
D
x
e
x
x
=
0
=
1
, so
a
1
=
1
.
Next:
We also want the polynomial to have the same curvature at zero as the
exponential function, so
′
′
P
(0)
=
2
a
2
+
3
⋅
2
a
3
x
x
=
0
=
2
a
2
=
D
xx
e
x
x
=
0
=
1
, so
a
2
=
1/2
.
Next:
We also want the polynomial to have the same change in curvature at zero as the
exponential function, so
′
′
′
P
(0)
=
3
⋅
2
a
3
=
D
xxx
e
x
x
=
0
=
1
, so
a
3
=
1/6
.
Hence, the appropriate polynomial would be
P
(
x
)
=
f
(0)
+
′
f
(0)
x
+
′
′
f
(0)
x
2
+
′
′
′
f
(0)
x
3
=
1
+
x
+
x
2
2
+
x
3
6
.
Definition
The
n
th Taylor polynomial
(centered at
a
) is
P
n
(
x
)
=
f
(
a
)
+
′
f
(
a
)(
x
−
a
)
+
′
′
f
(
a
)
(
x
−
a
)
2
2!
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 Spring '08
 Hodges
 Taylor Series, Dr. Berg, Summer ‘09

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