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9.7 Taylor Series
Brook Taylor1685 - 1731
Taylor Series
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 0.25 0.3
-0.3
-0.2
-0.1
0.1
0.2
0.3
x
y
Identify this graph:What if we zoom out a bit?And a bit more?
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
x
y
-8 -6 -4 -2 2 4 6 8
-4
-2
2
4
x
y
What the?! The actual function is:
!17!15!13!11!9!7!5!3)(
171513119753 xxxxxxxxxxf
Suppose we wanted to find a fifth degree polynomial of the form:
at 0x that approximates the behavior of
If we make , and the first, second, third, fourth, and fifth derivatives the same, then we would have a pretty good approximation.
0 0P f
55
44
33
2210)( xaxaxaxaxaaxP
xxf sin)(
)0(0 fP )0(0 fP )0(0 fP
)0(0
iv
fP iv
)0(0v
fP v
00P a
55
44
33
2210)( xaxaxaxaxaaxP xxf sin)(
xxf sin)( 00sin)0( f
55
44
33
2210)( xaxaxaxaxaaxP
0f00a
0P
xxf cos)( 10cos)0( f
45
34
2321 5432)( xaxaxaxaaxP
0f 1
1a 0P1)0( aP
xxf sin)( 00sin)0( f
35
2432 4534232)( xaxaxaaxP
0f 0
22a 0P 22)0( aP
02 a
55
44
33
2210)( xaxaxaxaxaaxP xxf sin)(
02 a
00P axxf sin)(
00sin)0( f
55
44
33
2210)( xaxaxaxaxaaxP
00a
xxf cos)( 10cos)0( f
45
34
2321 5432)( xaxaxaxaaxP
11a1)0( aP
xxf sin)( 00sin)0( f
35
2432 4534232)( xaxaxaaxP
022a22)0( aP
11 a
00 a
55
44
33
2210)( xaxaxaxaxaaxP xxf sin)(
02 a11 a00 a
xxf cos)( 10cos)0( f
0f -1323 a
0P
2543 3453423)( xaxaaxP
323)0( aP
!3
1
23
13
a
xxf iv sin)( 00sin)0( ivf
xaaxP iv54 2345234)(
0ivf04234 a
0ivP4234)0( aP iv
04 a
xxf cos)()5( 10cos)0()5( f
5)5( 2345)( axP
0)5(f1
5!5 a 0)5(P5
)5( !5)0( aP !5
15 a
55
44
33
2210)( xaxaxaxaxaaxP xxf sin)(
02 a11 a00 a
xxf cos)( 10cos)0( f -1323 a
2543 3453423)( xaxaaxP
323)0( aP
!3
1
23
13
a
xxf iv sin)( 00sin)0( ivf
xaaxP iv54 2345234)(
04234 a
4234)0( aP iv 04 a
xxf cos)()5( 10cos)0()5( f
5)5( 2345)( axP
15!5 a 5
)5( !5)0( aP !5
15 a
3a!3
1
55
44
33
2210)( xaxaxaxaxaaxP xxf sin)(
02 a11 a00 a 04 a!5
15 a3a
!3
1
)(xP 0a 1a 2a 3a 4a 5a 5x4x3x2xx
0 0 01
!3
1!5
1
53
!5
1
!3
11)( xxxxP
P(x) is the fifth degree polynomial thathas the same first through fifth derivatives at x = 0 that f(x) = sin x has. We call P(x) the 5th degree Taylor Polynomial of f(x) at x = 0.
2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
)1ln()( xxf
01ln)0( f 0)0( aP 00 a
1
1)(
xxf
110
1)0(
f
44
2321 432)( xaxaxaaxP
1)0( aP 11 a
2 3 40 1 2 3 4P x a a x a x a x a x
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
2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
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
1
1f x
x
10 1
1f
22 3 42 6 12P x a a x a x
20 2P a 2
1
2a
2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
2 3 41 2 60 1
2 6 24P x x x x x
2 3 4
02 3 4
x x xP x x ln 1f x x
P x
f x
If we plot both functions, we see that near zero the functions match very well!
This pattern occurs no matter what the original function was!
Our polynomial: 2 3 41 2 60 1
2 6 24x 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
This is called a Taylor Polynomial. If we continue this pattern infinitely, we get a Taylor Series. When theseries is centered at x = 0, it is called a Maclaurin Series.
If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:
Taylor Series:
(generated by f at )x a
2 3
2! 3!
f a f aP x f a f a x a x a x a
Maclaurin Series:
(generated by f at )
0x
...!3
)0(
!2
)0()0()0()( 32
x
fx
fxffxP
2 3 4 5 61 0 1 0 1
1 0 2! 3! 4! 5! 6!
x x x x xP x x
Example: Generate the Maclaurin Series for cosy x
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
1 2! 4! 6! 8! 10!
x x x x xP x
...!3
)0(
!2
)0()0()0()( 32
x
fx
fxffxP
The formula for Maclaurin Series is:
cosy x 2 4 6 8 10
1 2! 4! 6! 8! 10!
x x x x xP x
The more terms we add, the better our approximation.
example: 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 1
2f
cosf x x 02
f
sinf x x 12
f
4 cosf x x 4 0
2f
3 5
2 2
2 3! 5!
x xP x x
...!3
)()(
!2
)()())(()()(
axaf
axafaxafafxP
When referring to Taylor polynomials, we can talk about number of terms, order or degree.
2 4
cos 12! 4!
x xx This is a polynomial in 3 terms.
It is a 4th order Taylor polynomial, because it was found using the 4th derivative.
It is also a 4th degree polynomial, because x is raised to the 4th power.
The 3rd order polynomial for is , but it is degree 2.
cos x2
12!
x
The x3 term drops out when using the third derivative.
This is also the 2nd order polynomial.
A recent AP exam required the student to know the difference between order and degree.
A Taylor Polynomial is used to approximatea function.
!4!21cos
42 xxx
A Taylor Series is equal to a function.
...!5!4!2
1cos542
xxx
x
In practice, we usually use a Taylor Polynomialto approximate a function so we also want toknow the error in using the approximation.
!4!21cos
42 xxx
The error (also known as the remainder, Rn(x))is the difference between the actual value of the function, f(x), and the polynomial of degree n,Pn(x).
)()()( xRxPxf nn )(!4!2
1cos 4
42
xRxx
x )()()( xRxPxf nn
)()()( xRxPxf nn
Taylor’s Theorem specifies that the remainderRn(x) (also called the Lagrange Error or Lagrangeform of the remainder) is given by the formula:
1)1(
)()!1(
)()(
nn
n axn
zfxR
This looks a lot like the next term in the TaylorSeries except that it is not evaluated at a butat some number z which is between x and a.
z
z
)(!
)()(...
!2
)()())(()()(
2
xRn
axaf
axafaxafafxf n
nn
1)1(
)()!1(
)()(
nn
n axn
zfxR
)(!
)()(...
!2
)()())(()()(
2
xRn
axaf
axafaxafafxf n
nn
Taylor’s Theorem guarantees that there exists a value of z between a and x such that:
)()()( xRxPxf nn
It usually isn’t possible to find the value of but you can often find an upper bound on it.This will be called max
zf n 1
zf n 1
55
4 )0(!5
)()( x
zfxR
)(!4!2
1cos 4
42
xRxx
x Example:
Therefore the Lagrange error term
zzfxxf sin)( then ,cos)( Since 5
zsin of value theknowt don' We 1sin1 that know do but we z
1)( that meanswhich 5 zf
555
4 !5
1)0(
!5
)()( xx
zfxR
1)1(
)()!1(
)()(
nn
n axn
zfxR
