Math 31B. Spring ’14.
WEEK 7.
Part
1. Taylor’s theorem: It gives a way of approxmating a function by a polynomial
f
(
x
) =
f
(
a
) +
f
0
(
a
)(
x

a
) +
1
2
f
00
(
a
)(
x

a
)
2
+
. . .
+
1
k
!
f
(
k
)
(
a
)(
x

a
)
k
+
. . .
+
1
N
!
f
(
N
)
(
a
)(
x

a
)
N
+
R
N
(
f, x, a
)
Here
f
(
k
)
(
a
) represents the value at
a
of the
k
th derivative of
f
. The quantity
R
N
(
f, x, a
) is
very small
in the sense that it approximates zero faster than (
x

a
)
N
, in the sense that
lim
x
!
a
R
N
(
f, x, a
)
(
x

a
)
N
= 0
.
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Expert Verified
Since
f
0
(
x
) =
e
x
, then
f
(
k
)
(
x
) =
e
x
always, thus
f
(
k
)
(0) = 1 for
k
= 1
,
2
,
3
, . . .
. Using the definition of