Taylor series in 1 and 2 variable

TITLE OUTLINE
 INTRODUCTION
 HISTORY
 TAYLOR SERIES
 MACLAURIAN SERIES
 EXAMPLE

Brook Taylor
1685 - 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

Suppose we wanted to find a fourth degree polynomial of
the form:
  2 3 4
0 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


  2 3 4
0 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 4
0 1 2 3 4P x a a x a x a x a x    
  00P a 0 0a 
 
1
1
f x
x
 

 
1
0 1
1
f   
  2 3
1 2 3 42 3 4P x a a x a x a x    
  10P a  1 1a 
 
 
2
1
1
f x
x
  

 
1
0 1
1
f     
  2
2 3 42 6 12P x a a x a x   
  20 2P a  2
1
2
a  


  2 3 4
 
 
3
1
2
1
f x
x
  

 0 2f  
  3 46 24P x a a x  
  30 6P a  3
2
6
a 
 
 
 
4
4
1
6
1
f x
x
 

 
 4
0 6f  
 
 4
424P x a
 
 4
40 24P a 4
6
24
a  
 
 
2
1
1
f x
x
  

 
1
0 1
1
f     
  2
2 3 42 6 12P x a a x a x   
  20 2P a  2
1
2
a  


  2 3 4
  2 3 41 2 6
0 1
2 6 24
P x x x x x    
 
2 3 4
0
2 3 4
x x x
P x x        ln 1f x x 
-1
-0.5
0
0.5
1
-1 -0.5 0.5 1
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
 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 6
0 1
2 6 24
x x x x   
has the form:    
     
 4
2 3 40 0 0
0 0
2 6 24
f f f
f f x x x x
 
   
or:          
 4
2 3 40 0 0 0 0
0! 1! 2! 3! 4!
f f f f f
x x x x
  
   


Maclaurin Series:
(generated by f at )0x 
     
   2 30 0
0 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
 
        


 
2 3 4 5 6
1 0 1 0 1
1 0
2! 3! 4! 5! 6!
x x x x x
P 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 x
P x       


-1
0
1
-5 -4 -3 -2 -1 1 2 3 4 5
cosy x  
2 4 6 8 10
1
2! 4! 6! 8! 10!
x x x x x
P x       
The more terms we add, the better our approximation.

Hint: On the TI-89, the factorial symbol is: 

example:  cos 2y x
Rather than start from scratch, we can use the function
that we already know:
 
         
2 4 6 8 10
2 2 2 2 2
1
2! 4! 6! 8! 10!
x x x x x
P x       


example:  cos at
2
y x x

 
 
2 3
0 1
0 1
2 2! 2 3! 2
P x x x x
       
             
     
  cosf x x 0
2
f
 
 
 
  sinf x x   1
2
f
    
 
  cosf x x   0
2
f
   
 
  sinf x x  1
2
f
   
 
 
 4
cosf x x
 4
0
2
f
 
 
 
 
3 5
2 2
2 3! 5!
x x
P x x
 

   
    
            
 


There are some Maclaurin series that occur often enough
that they should be memorized. They are on your
formula sheet.


When referring to Taylor polynomials, we can talk about
number of terms, order or degree.
2 4
cos 1
2! 4!
x x
x    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 x
2
1
2!
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.


The TI-89 finds Taylor Polynomials:
taylor (expression, variable, order, [point])
  cos , ,6x xtaylor
6 4 2
1
720 24 2
x x x
  
  cos 2 , ,6x xtaylor
6 4
24 2
2 1
45 3
x x
x

  
  cos , ,5, / 2x x taylor    
5 3
2 2 2
3840 48 2
x x x     
 

9F3

Taylor series in 1 and 2 variable

Taylor series in 1 and 2 variable

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Taylor series in 1 and 2 variable