Calculus and Elementary Analysis(I)
Instructor: You Xiong
Teaching Assistant: Wei Ming
2007.9-2009.1
1

Objectives
• 1.Understand the Taylor Theorem with
Peano’s and Lagrangian remainders
• 2.Be able to find the n-th Taylor polynomials of simple functions
• 3.Be able to apply Taylor’s theorem to complicated limits
2

Motivation
• Recall: the geometrical idea of the tangent line of a differentia function
3

For a function f ( x ) that is differentiable at x=x
0
, the tangent is a close approximation of the function in a neighborhood of the tangent point x
0
.
y y
  
0

'( )(
0 x
 x
0
)
    
0 x
 x
0
4
We call the equation of the tangent the linearization of the function.
x


Ambition:
Approximating of differential functions more precisely by polynomials of higher degrees.
5

Taylor Polynomials
•Definition.
The Taylor polynomial of degree n for a function f ( x ) which is n times continuously differentiable at x
0 is
T n
(x)=
(
0
)

'(
0
)( x
 x
0
)
 f ''( x
0
)
2!
( x
 x
0
)
2 
...
 f ( x
0
) n !
( x
 x
0
) n
6

Approximation: f x

T x n

R x
( )

T x n
( )
R ( x )
7

Taylor’s Theorem
For a function f ( x ) which is n times continuously differentiable at x
0
, if T
polynomial of f at x
0 n
(x) is the Taylor
, then the remainder
( )

(( x
 x
0 n
) ) where
R n
(x) is called
Peano’s remainder
.
•Proof. (Using L’Hospital’s Rule for limits.)
8

Example1:
Find the Taylor formula for f x
 e x
, at x
0

0.
Solution.
Step1. Find f ( )

 
( ) 
, then f (0)
 k

0,1, 2, .
Step2. Form the Taylor polynomial n
( )
 f (0)
 f '(0) x
9
 f ''(0)
2!
x
2   f n !
(0) x n
.

Example1:
Find the Taylor formula for f x
 e x
, at x
0

0.
Solution.
Step3. Add the remainder R n give the Taylor formula.
(x) to
( )
 f (0)
 f '(0) x
 f ''(0)
2!
x
2   f n !
(0) x n  x n
( ).
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Example1:
Find the Taylor formula for f x
 e x
, at x
0

0.
Solution.
e x
1 x 1
2!
x
2  
1 n !
x n  x n
( ).
•Maclaurin’s formula .
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A Taylor Polynomial f ( a )
 f ' ( a )( x
 a )
 f '' ( a )
( x
 a )
2
2 !

...
 f
( n )
( a )
( x
 a ) n n !
is a (finite) sum with a defined degree.
12

A Taylor Series f ( a )
 f ' ( a )( x
 a )
 f '' ( a )
2 !
( x
 a )
2  f
( 3 )
( a )
3 !
( x
 a )
3 
...
is an infinite sum, i.e., a sequence of partial sums.
13

Sequence of Partial Sums
Each member of the sequence of partial sums is itself a Taylor polynomial.
A Taylor series is a sequence of Taylor polynomials .
14

Sequence of Taylor Polynomials
Let
15

As the Taylor Series
We actually define the Taylor series for at to be the sequence
16

An Example
The first few Taylor polynomials for at
The Taylor series:
17

Assumptions
For a Taylor polynomial, you need derivatives up to order
For a Taylor series, you need derivatives of all orders .
18

Examples
We could find Taylor series for at at at at
19

Notation
Using sigma notation write k



0 f
( k ) k !
( a )
( x
 a ) k  f ( a )
 f ' ( a )( x
 a )
 f '' ( a )
2 !
( x
 a )
2  f
( 3 )
( a )
3 !
( x
 a )
3 
...
for the Taylor series for
20 at

Terminology
The Taylor series for at is called the Maclaurin series for
21

Will the Calculator Find a Taylor Series?
Maybe; try it for at
22

Where Does One Start?
Start with a Taylor polynomial.
Try using a large degree.
Can you see the pattern?
I can’t!
23

Now What?
Find the coefficient of the degree n term in the Taylor series for at
Start by finding the n th derivative of
24

Use the Calculator to...
get successive expressions for
25

The Pattern?
Look at a couple more:
26

Conclusion is good for n

2 , and then
27

When a

4 ,
28

Thus
The coefficient for the degree n term is in the Taylor series for at
29

Notice
Finding a Taylor series means finding the coefficients.
30

Taylor Series f ( a )
 f ' ( a )( x
 a )
 f '' ( a )
2 !
( x
 a )
2  f
( 3 )
( a )
3 !
( x
 a )
3 
...
Degree n term
Coefficient of Degree n Term
31

Separate Issues
• What is the Taylor series for a function at a point?
• For what values of x does the Taylor series for a function converge ?
• Does the Taylor series for a function converge to that function ?
32

A Weird Example
Let
It’s hard to see that it even exists, but
Thus the coefficients in the Taylor series for at are all
33

Meaning...
This Taylor series describes the function well but only at one point, 0.
In cases like this, Taylor series aren’t good for much.
34

Lucky for Us
All of the usual suspects can be well represented by their Taylor series at all points where they are infinitely differentiable.
The Taylor series for all of our favorite functions converge to the functions at least on a decent sized interval, if not on the entire real line.
35

Meaning...
• For all algebraic functions,
• for trigonometric functions and their inverses,
• for exponential functions and logarithms, there are excellent polynomial approximations to the functions on intervals surrounding points where the functions have derivatives of all orders.
36

Example
Find a polynomial approximation to that is to within two decimal places of on the interval (2,6).
Solution?
Use a Taylor polynomial at
37

What Degree Do We Need?
Use your technology to figure it out!
38

How Can You Tell?
If the polynomial and the function agree at the endpoints, they agree at all the points between.
That’s a Big Theorem .
39

Power Series
A power series at a is a function of the form p ( x )
 a
0
 a
1
( x
 a )
 a
2
( x
 a )
2  a
3
( x
 a )
3 
...
The a i s are the coefficients .
40

Example
1
 x
 x
2  x
3  x
4
...
is a power series at
All the coefficients are 1.
41

Recall
1
 x
 x
2  x
3  x
4
...

1
1
 x
, for 1
 x

1 .
42

Huge Theorem
If for some positive number r , then is the Taylor series for f at a.
43

So...
1
 x
 x
2  x
3  x
4
...
is the Maclaurin series for f ( x )

1
1
 x
.
44

More Interesting Stuff
If a power series converges on an interval, we can
• differentiate term by term to get another convergent power series
• integrate term by term to get another convergent power series
• take limits term by term, on the interval of convergence
• do arithmetic term by term to get still more convergent power series.
45

Example
In other words, for a given value of x converges to
46

True For All x...
means we can substitute for
That gives us which is the Maclaurin series for that function.
47

More Cute Tricks
1
 x
 x
2  x
3  x
4
...

1
1
 x
, for 1
 x

1 .
means
1

1 x
2

1

( x
2
)

( x
2
)
2
48

( x
2
)
3 
...

One Last Example
Find the Maclaurin series for x sin( x ).
Solution
Start with the Maclaurin series for sin( x ).
It converges to sin( x )
 x .
49

Thus...
x sin( x )
 x ( x
 x
3
3 !
 x
5
5 !
 x
7
7 !

...)
 x
2  x
4
3 !
 x
6
5 !
 x
8
7 !

...
This is the Maclaurin series for x sin( x ).
It converges to the function on the whole real line.
50

Summary:
Memorize the following Taylor formulas:
• The Maclaurin series for
• e x =
• sinx=
• Cosx=
• ln(1+x)=
• (1+x) a =
• 1/(1-x)=
51

Written Work:
P141.
1(1),(2)
2(1)(3)
3(2)
P142.
5(1)
52

Computer Work:
P155.
4
Draw the graph of the function f(x)=xe -x , and its first five Taylor Polynomial approximations at x
0
=0.
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