taylor series in 1 and 2 variable
TRANSCRIPT
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BRANCH :CIVIL-2
Taylor Series For One And Two Variable
By Rajesh Goswami
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TITLE OUTLINE
INTRODUCTION HISTORYTAYLOR SERIESMACLAURIAN SERIESEXAMPLE
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Brook Taylor1685 - 1731
9.2: 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.
Greg Kelly, Hanford High School, Richland, Washington
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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 , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.
0 0P f
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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
11
f xx
10 11
f
2 31 2 3 42 3 4P x a a x a x a x
10P a 1 1a
2
11
f xx
10 11
f
22 3 42 6 12P x a a x a x
20 2P a 212
a
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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
3
121
f xx
0 2f
3 46 24P x a a x
30 6P a 326
a
44
161
f xx
4 0 6f
4424P x a
440 24P a 4
624
a
2
11
f xx
10 11
f
22 3 42 6 12P x a a x a x
20 2P a 212
a
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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 12 6 24
P x x x x x
2 3 4
02 3 4x 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!
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This pattern occurs no matter what the original function was!
Our polynomial: 2 3 41 2 60 12 6 24
x x x x
has the form: 42 3 40 0 0
0 02 6 24f f f
f f x x x x
or: 42 3 40 0 0 0 0
0! 1! 2! 3! 4!f f f f f
x x x x
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Maclaurin Series:
(generated by f at )0x
2 30 00 0
2! 3!f f
P x f f x x x
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 a
P x f a f a x a x a x a
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2 3 4 5 61 0 1 0 11 0
2! 3! 4! 5! 6!x x x x xP x x
example: 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
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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.
Hint: On the TI-89, the factorial symbol is:
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example: cos 2y x
Rather than start from scratch, we can use the function that we already know:
2 4 6 8 102 2 2 2 21
2! 4! 6! 8! 10!x x x x x
P x
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example: cos at 2
y x x
2 30 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
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There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet.
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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.
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The TI-89 finds Taylor Polynomials:
taylor (expression, variable, order, [point])
cos , ,6x xtaylor6 4 2
1720 24 2x x x
cos 2 , ,6x xtaylor6 4
24 2 2 145 3x x x
cos , ,5, / 2x x taylor 5 32 2 23840 48 2x x x
9F3
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Thank You