Brook Taylor developed ideas regarding infinite series in the late 17th century. The document discusses Taylor series and Maclaurian series, which are ways of approximating functions using polynomials. Taylor series expand a function around a point x=a using derivatives, while Maclaurian series use a point of x=0. Examples are given of approximating common trigonometric functions like cosine using Taylor polynomials of increasing degrees. The order, degree, and number of terms in the approximations are defined.

1. BRANCH :CIVIL-2

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

3. 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

4. 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


5.   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  


6.   2 3 4
0 1 2 3 4P x a a x a x a x a x        ln 1f x x 
 
 
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  


7.   2 3 4
0 1 2 3 4P x a a x a x a x a x        ln 1f x x 
  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!


8. 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
  
   


9. 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
 
        


10.  
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       


11. -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: 

12. 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       


13. 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
 

   
    
            
 


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

15. 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.


16. 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