9.3
Taylor’s Theorem
Quick Review
Find the smallest number M that bounds f from above on the
interval I (that is, find the smallest M such that f ( x)  M for
all x in I ).
1. f ( x)  2 cos(3 x), I   -2 , 2 
2. f ( x)  x  3 I  1, 2
2
3. f ( x)  2
x
I   -3, 0
x
4. f ( x) 
I   -2, 2
x 1
2
2
7
1
1
2
Quick Review
Tell whether the function has derivatives of all orders at the
of
derivatives
has
function
the
whether
Tell
given values of a.
x
, a0
5.
x 1
6. x  4 , a  2
2
7. sin x  cos x, a  
8. e , a  0
-x
3
2
9. x , a  0
Yes
No
Yes
Yes
No
all orde
What you’ll learn about




Taylor Polynomials
The Remainder
Remainder Estimation Theorem
Euler’s Formula
Essential Questions
How do we determine the error in the approximation
of a function represented by a power series by its
Taylor polynomials?
Example Approximating a Function to
Specifications
1. Find a Taylor polynomial that will serve as an adequate substitute for
sin x on the interval [– , ].
Choose Pn(x) so that |Pn(x) – sin x| < 0.0001 for every x in the interval [– , ].
We need to make |Pn() – sin  | < 0.0001, because then Pn then will be
adequate throughout the interval
Pn    sin   0.0001
Pn    0.0001


3
3!


5
5!


7
7!
Evaluate partial sums at x = , adding one term
at a time.


9


11


13
9 ! 11! 13 !
 2.114256749 10
5
Taylor’s Theorem with Remainder
Let f has a derivative of all orders in an open interval I containing a,
then for each positive integer n and for each x in I
f a 
f n  a 
2
x  a    
x  a n  Rn x ,
f x   f a   f a x  a  
2!
n!
f n1 c 
n 1
x  a  for some c between a and x.
where Rn x  
n  1!
Example Proving Convergence of a
Maclaurin Series
2 k 1

x
k
2. Prove that the series   1
converges to sin x for all real x.
2k  1!
k 0
n 1
Consider Rn(x) as n → ∞. By Taylor’s Theorem, Rn x  
c  x  a n1
n  1!
f
where f (n+1)(c) is the (n + 1)st derivative of sin x evaluated at some c between x and 0.
Since,  1  f n1 c   1
n 1
n 1
x
1

f
c
n 1
n 1

x

x  0
Rn x  
n  1!
n  1!
n  1!
As n → ∞, the factorial growth is larger in the bottom than the exp. growth in the top.
Therefore, as n  
x n1
n  1!
 0 for all x.
This means that Rn(x) → 0 for all x.
Remainder Estimation Theorem
If there are positive constants M and r such that f n 1 t   Mr n 1
for all t between a and x, then the remainder Rn(x) in Taylor’s Theorem
satisfies the inequality
Rn x   M
r
n 1
xa
n  1!
n 1
.
If these conditions hold for every n and all the other conditions of
Taylor’s Theorem are satisfied by f , then the series converges to f (x).
Example Proving Convergence
3. Use the Remainder Estimation Theorem to prove the following for
k

all real x.
x
x
e 
k 0
k!
We have already shown this to be the Taylor series generated by e x at x = 0.
We must verify Rn(x) → 0 for all x.
To do this we must find M and r such that
f n1 t   et is bounded by Mr n1 for t between 0 and arbitrary x.
Let M be the maximum value for e t and let r = 0.
If the interval is [0, x ], let M = e x .
If the interval is [x, 0 ], let M = e 0 = 1.
In either case, e x < M throughout the interval, and the Remainder Estimation
Theorem guarantees convergence.
Euler’s Formula
e  cos x  i sin x
ix
Quick Quiz Sections 9.1-9.3

1. Which of the following is the sum of the series 
n 0
(A)
e
e -

 -e

(C)
 -e
(B)
2
2
e
(D)
e -
(E) The series diverges
2

e
n
2n
?
Quick Quiz Sections 9.1-9.3

1. Which of the following is the sum of the series 
n 0
(A)
e
e -

 -e

(C)
 -e
(B)
2
2
e
(D)
e -
(E) The series diverges
2

e
n
2n
?
Quick Quiz Sections 9.1-9.3
2. Assume that f has derivatives of all orders for all real numbers x,
f (0)  2, f '(0)  -1, f ''(0)  6, and f '''(0)  12. Which of the following
is the third order Taylor polynomial for f at x  0?
(A) 2  x  3x  2 x
(B) 2  x  6 x  12 x
1
(C) 2  x  3 x  2 x
2
(D)  2  x  3 x  2 x
(E) 2  x  6 x
2
3
2
3
2
3
2
2
3
Quick Quiz Sections 9.1-9.3
2. Assume that f has derivatives of all orders for all real numbers x,
f (0)  2, f '(0)  -1, f ''(0)  6, and f '''(0)  12. Which of the following
is the third order Taylor polynomial for f at x  0?
(A) 2  x  3 x  2 x
(B) 2  x  6 x  12 x
1
(C) 2  x  3 x  2 x
2
(D)  2  x  3 x  2 x
(E) 2  x  6 x
2
3
2
3
2
3
2
2
3
Quick Quiz Sections 9.1-9.3
3. Which of the following is the Taylor series generated by
f ( x)  1/ x at x  1?
(A)   x  1

n
n 0
(B)   1 x

n
n
n 0
(C)   1  x  1

n
n
n 0
(D)   1

n
 x  1
n!
(E)   1  x  1
n
n 0

n 0
n
n
Quick Quiz Sections 9.1-9.3
3. Which of the following is the Taylor series generated by
f ( x)  1/ x at x  1?
(A)   x  1

n
n 0
(B)   1 x

n
n
n 0
(C)   1

n
n 0
(D)   1

n
 x  1
n
 x  1
n!
(E)   1  x  1
n
n 0

n 0
n
n
Pg. 386, 7.1 #1-25 odd