Warm up Find the Maclaurin Series for f x x ( ) cos fileFind the power series representation of f x...
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Warm up Find the Maclaurin Series for () cos fx x
Transcript of Warm up Find the Maclaurin Series for f x x ( ) cos fileFind the power series representation of f x...
Warm up
Find the Maclaurin Series for ( ) cosf x x
Agenda
-Warm up
-9. 10 Notes
-HW p.673;1-11 odd,27-
39odd,41,47-
51odd,53,54,63-67odd
DESK
Notes/warm
up
Class
work/Homework
LT 1; 9.10 Derive the Taylor Series of a function using a list of known power series for elementary function.
Find the power series representation of
( ) arctan ,center at 0f x x
Know: 2
1'( )
1f x
x
2
1
1 x 0n
2
1
1 x
2n
x 2
0
1n n
n
x
2
1
1 x 2
0
1n n
n
x
arctan x 2 1
0
1
2 1
n n
n
xc
n
Let x = 0 C=0
2 1
0
1
2 1
n n
n
x
n
Want: 1
1 x
9.10 Taylor and Maclaurin Series
Common Maclaurin Series (pg 670):
1
1 x
2 31 ...x x x 0n
nx ( 1,1)
1
1 x
2 31 ...x x x 0n
1n nx ( 1,1)
1
x 2 31 ( 1) ( 1) ( 1) ...x x x
0n
1 1n n
x
(0,2)
Common Maclaurin Series (pg 670):
ln x 2 3 4( 1) ( 1) ( 1)
( 1) ...2 3 4
x x xx
1n
1
1 1n n
x
n
(0,2]
xe 2 3
1 ...2! 3!
x xx
0n
!
nx
n( , )
sin x 3 5 7
...3! 5! 7!
x x xx
0n
2 11
2 1 !
n nx
n
( , )
cos x 2 4 6
1 ...2! 4! 6!
x x x
0n
21
2 !
n nx
n
( , )
Common Maclaurin Series (pg 670):
arctan x 3 5 7
...3 5 7
x x xx
0n
2 11
2 1
n nx
n
[ 1,1]
arcsin x 3 5 71 3 1 3 5
...2 3 2 4 5 2 4 6 7
x x xx
0n
2 1
2
2 !
2 ! 2 1
n n
n
n x
n n
[ 1,1]
Ex : find the power series for
( ) cosf x x
cos x 2 4 6
1 ...2! 4! 6!
x x x
0n
21
2 !
n nx
n
( , )
2 4 6
1 ...2! 4! 6!
x x x
0n
2
1
2 !
nnx
n
2 3
1 ...2! 4! 6!
x x x
0n
1
2 !
n nx
n
( , )
cos x
Ex : find the maclaurin series for
2( ) sin 2f x x x
sin x 3 5 7
...3! 5! 7!
x x xx
0n
2 11
2 1 !
n nx
n
( , )
sin 2x
3 5 72 2 2
2 ...3! 5! 7!
x x xx
0n
2 11 2
2 1 !
n nx
n
2 sin 2x x
3 5 7
22 2 2
2 ...3! 5! 7!
x x xx x
( , )
0
n
2 11 2
2 1 !
n nx
n
2x
3 3 5 5 7 72 2 2 2
2 ...3! 5! 7!
x x xx x
0
n
2 1 2 11 2
2 1 !
n n nx
n
2x
3 5 5 7 7 93 2 2 2
2 ...3! 5! 7!
x x xx
0n
2 1 2 31 2
2 1 !
n n nx
n
Ex : Find the Taylor Series for 𝑓 𝑥 =arctan(𝑥2)
𝑥𝑐𝑒𝑛𝑡𝑒𝑟𝑎𝑡
π
2
9.10 Day 2: (1)Power Series Multiplication/ Division. (2)Using trig identity to help with creating power series (3) Approximating with power series
Ex : find the 1st 3 none zero terms of maclaurin series for 2
( ) cosxf x e xmultiplication of power series
cos x 2 4 6
1 ...2! 4! 6!
x x x
2xe
2 32 2
21 ...2! 3!
x xx
1 2x
4
2
x
6
...6
x
Ex : find the 1st 3 none zero terms of maclaurin series for 2
( ) cosxf x e xmultiplication of power series
cos x2 4 6
1 ...2 24 720
x x x
2xe1 2x
4
2
x
6
...6
x
1 2x
4
2
x
6
...6
x
2
2
x
4
2
x
6
4
x
4
24
x6
24
x
123
2x 425
24x
Ex : find the 1st four none-zero terms of maclaurin series for
( ) arctanxf x e xmultiplication of power series
xe
arctan xx
x3
3
x
5
5
x
7
...7
x
2x4
3
x
6
5
x
3
2
x 5
6
x
x 2x3
6
x
2 3
1 ...2! 3!
x xx
3 5 7
...3 5 7
x x xx
8
...7
x
7
...10
x
4
6
x
4
6
x
Division of power series Ex : find the 1st three none-zero terms of maclaurin series for
( )sin
xf x
x sin x
3 5 7
...3! 5! 7!
x x xx
3 5
...6 120
x xx x
1
x-3
6
x
5
120
x
3
6
x5
120
x
2
6
x
3
6
x 5
36
x-
57
360
x
47
360
x
Division of power series
Ex : find the 1st three none-zero terms of maclaurin series for sin
( ) tancos
xf x x
x
sin x 3 5 7
...3! 5! 7!
x x xx
2 4
1 ...2 24
x x
3 5
-... 6 120
x xx
cos x 2 4 6
1 ...2! 4! 6!
x x x
x
x3
2
x
5
24
x-
3
3
x 5
30
x
3
3
x
3
3
x 5
6
x-
52
15
x
52
15
x
Ex : find the 1st three non zero term for
2( ) sinf x x
sin x 3 5 7
...3! 5! 7!
x x xx
multiplication of power series
3 5
...6 120
x xx
3 5
...6 120
x xx
2x4
6
x
6
120
x
4
6
x
6
36
x
6
120
x+
2x4
3
x
62
45
x
Ex : find the 1st three non zero term for
2( ) sinf x x
Using Trig Identity
2 1 1sin cos 2
2 2x x
cos x 2 4 6
1 ...2! 4! 6!
x x x
cos 2x
2 4 62 2 2
1 ...2! 4! 6!
x x x
1
cos 22
x
2 4 62 2 21
...2 2 2! 2 4! 2 6!
x x x
1 1
cos 22 2
x
2 4 62 2 21 1
...2 2 2 2! 2 4! 2 6!
x x x
1 1
cos 22 2
x
2 22
4
x4 42
48
x
6 62
1440
x
2x4
3
x
62
45
x
1n
1
1 1n n
x
n
2 3 4( 1) ( 1) ( 1)
( 1) ...2 3 4
x x xx
ln x
(0,2]
2 3 4
...2 3 4
x x xx ln( 1)x
3 4 52 ...
2 3 4
x x xx ln( 1)x x
3 4 5 6
...3 4 2 5 3 6 4
x x x x
1/4
0
ln( 1)x x 1
10,000
Solution :
5
1/ 4
5 3.000065 .0001
3 4 5 6
...3 4 2 5 3 6 4
x x x x
1/4
0
ln( 1)x x 1
10,000
