9.7 and 9.10 Taylor Polynomials and Taylor Series
Jan 03, 2016
Suppose we wanted to find a fourth degree polynomial of the form:
2 3 40 1 2 3 4P x a a x a x a x a x
ln 1f x x at 0x that approximates the behavior of
If we make f (0) = P(0), f ’(0) = P’(0), f ’’(0) = P’’(0), and so on, then we would have a pretty good approximation.
2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
ln 1f x x
0 ln 1 0f
2 3 40 1 2 3 4P x a a x a x a x a x
00P a 0 0a
1
1f x
x
10 1
1f
2 31 2 3 42 3 4P x a a x a x a x
10P a 1 1a
2
1
1f x
x
10 1
1f
22 3 42 6 12P x a a x a x
20 2P a 2
1
2a
3
12
1f x
x
0 2f
3 46 24P x a a x
30 6P a 3
2
6a
4
4
16
1f x
x
4 0 6f
4424P x a
440 24P a 4
6
24a
2 3 41 2 60 1
2 6 24P x x x x x
This is called the Taylor Polynomial of degree 4.
ln 1x 2 3 4
02 3 4
x x xP x x
Our polynomial 2 3 41 2 6
0 12 6 24
x x x x
has the form: 42 3 40 0 0
0 02 6 24
f f ff f x x x x
or: 42 3 40 0 0 0 0
0! 1! 2! 3! 4!
f f f f fx x x x
P x
f x
If we plot both functions, we see that near zero the functions match very well!
Definition
The series of the form
2 3
1! 2! 3!
f c f c f cf c x c x c x c
Theorem
If f (x) is represented by a power series
for all x in an open interval I containing c, then( ) ( )
!
n
n
f ca
n
is called the Taylor Series for f (x) at c. If c = 0, then the series is called the Maclaurin series for f .
0
( )nn
n
a x c
( )
0
( )( )
!
nn
n
f cx c
n
cosf x x 0 1f
sinf x x 0 0f
cosf x x 0 1f
sinf x x 0 0f
4 cosf x x 4 0 1f
2 4 6 8 10
cos 1 2! 4! 6! 8! 10!
x x x x xx
2 3 4cos2! 3!
1 0
4!
11 0x x x x x
Example
( ) cosf x xFind the Maclaurin series for
( ) cos 2f x x
Rather than start from scratch, we can use the function that we already know:
2 4 6 8 102 2 2 2 2
cos 2 1 2! 4! 6! 8! 10!
x x x x xx
Example
Find the Maclaurin series for
2 4 6 8 10
cos 1 2! 4! 6! 8! 10!
x x x x xx
cos at 2
y x x
2 3
0 10 1
2 2! 2 3! 2P x x x x
cosf x x 02
f
sinf x x 12
f
cosf x x 02
f
sinf x x 12
f
4 cosf x x 4 02
f
3 5
2 2
2 3! 5!
x xP x x
ExampleFind the Taylor series for
sin x
cos x
sin x
cos x
sin x
0
1
0
1
0
0nf nf x
2 3 4sin2
0 1 00 1
! 3! 4!x x x x x
3 5 7
sin 3! 5! 7!
x x xx x
Example
Find the Maclaurin series for ( ) sinf x x
2 30 00 0
2! 3!
f fP x f f x x x
1( )
1f x
x
11 x
21 x
32 1 x
46 1 x
524 1 x
1
1
2
6 3!
24 4!
0nf nf x
2 3 42 31
1 2! 3!
! 4!1
!1
4x x x x
x
2 3 411
1x x x x
x
This is a geometric series witha = 1 and r = x.
Example
Find the Maclaurin series for
2 30 00 0
2! 3!
f fP x f f x x x
( ) ln 1f x x
ln 1 x
11 x
21 x
32 1 x
46 1 x
0
1
1
2
6 3!
0nf nf x
2 3 4ln 12
1
! 3! 4
20 1
!
3!x x x x x
2 3 4
ln 12 3 4
x x xx x
Example
Find the Maclaurin series for
2 30 00 0
2! 3!
f fP x f f x x x
( ) xf x e
xe
xe
xe
xe
xe
1
1
1
1
1
0nf nf x
2 3 41 1 11 1
2! 3! 4!xe x x x x
2 3 4
12! 3! 4!
x x x xe x
Example
Find the Maclaurin series for
2 3 4
12! 3! 4!
x x x xe x
An amazing use for infinite series:
Substitute xi for x.
2 3 4 5 6
1 2! 3! 4! 5! 6!
xi xi xi xi xi xie xi
2 2 3 3 4 4 5 5 6 6
1 2! 3! 4! 5! 6!
xi x i x i x i x i x ie xi
2 3 4 5 6
1 2! 3! 4! 5! 6!
xi x x i x x i xe xi
2 4 6 3 5
1 2! 4! 6! 3! 5!
xi x x x x xe i x
Factor out the i terms.
Euler’s Formula
2 4 6 3 5
1 2! 4! 6! 3! 5!
xi x x x x xe i x
This is the series for cosine.
This is the series for sine.
cos sinxie ix x Let x
cos sinie i
1 0ie i
1 0ie This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated.
Taylor series are used to estimate the value of functions (at least theoretically - nowadays we can usually use the calculator or computer to calculate directly.)
An estimate is only useful if we have an idea of how accurate the estimate is.
When we use part of a Taylor series to estimate the value of a function, the end of the series that we do not use is called the remainder. If we know the size of the remainder, then we know how close our estimate is.
Ex: Use to approximate over .2 4 61 x x x 2
1
1 x 1,1
the truncation error is , which is .8 10 12 x x x 8
21
x
x
When you “truncate” a number, you drop off the end.
Taylor’s Theorem with Remainder
If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x
in I:
2
2! !
nn
n
f fa af f f Rx a a x a x a x a xn
Lagrange Form of the Remainder
11
1 !
nn
n
f cR x x a
n
Remainder after partial sum Sn
where c is between
a and x.
This is also called the remainder of order n or the error term.
This is called Taylor’s Inequality.
Taylor’s Inequality
Lagrange Form of the Remainder
11
1 !
nn
n
f cR x x a
n
If M is the maximum value of on the interval
between a and x, then:
1nf x
1
1 !n
n
MR x x a
n
Note that this looks just
like the next term in the
series, but “a” has been
replaced by the number
“c” in . 1nf c
Find the Lagrange Error Bound when is used
to approximate and .
2
2
xx
ln 1 x 0.1x ln 1f x x 1
1f x x
21f x x
32 1f x x
22
0 00
1 2!
f ff f x x Rx x
Remainder after 2nd order term
2
202
xf x Rx x
On the interval , decreases, so
its maximum value occurs at the left end-point.
.1,.1 3
2
1 x
3
2
1 .1M
3
2
.9 2.74348422497
32.7435 .1
3!nR x 0.000457
Lagrange Error Bound
x ln 1 x2
2
xx error
.1 .0953102 .095 .000310
.1 .1053605 .105 .000361