9.3 taylor’s theorem quick review tell whether the function has derivatives of all orders at the...
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9.3
Taylor’s Theorem
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Quick Review
2
Find the smallest number that bounds from above on the
interval (that is, find the smallest such that ( ) for
all in ).
1. ( ) 2cos(3 ), -2 , 2
2. ( ) 3 1, 2
3. ( ) 2 -3,0
4.
x
M f
I M f x M
x I
f x x I
f x x I
f x I
2
( ) -2, 21
xf x I
x
2
71
2
1
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Quick Review
2
-
3
2
Tell whether the function has derivatives of all orders at the given values of .
5. , 01
6. 4 , 2
7. sin cos ,
8. , 0
9. , 0
x
a
xa
xx a
x x a
e a
x a
Tell whether the function has derivatives of all orders at the given values of a.
Yes
No
Yes
Yes
No
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What you’ll learn about Taylor Polynomials The Remainder Remainder Estimation Theorem Euler’s Formula
Essential QuestionsHow do we determine the error in the approximationof a function represented by a power series by itsTaylor polynomials?
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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
0001.0sin nP
0001.0nP
Evaluate partial sums at x = , adding one term at a time.
! 3
3
! 5
5
! 7
7
! 9
9
! 11
11
! 31
13 510114256749.2
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Taylor’s Theorem with RemainderLet 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
. and between somefor ! 1
where 11
xacaxn
cfxR n
n
n
,
! ! 2 2 xRax
n
afax
afaxafafxf n
nn
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Example Proving Convergence of a Maclaurin Series
. real allfor sin toconverges ! 12
1 series that theProve 2.0
12
xxk
x
k
kk
Consider Rn(x) as n → ∞. By Taylor’s Theorem,
where f (n+1)(c) is the (n + 1)st derivative of sin x evaluated at some c between x and 0.
11
! 1
nn
n axn
cfxR
11 Since, 1 cf n
11
0 ! 1
nn
n xn
cfxR
1
! 1
1
nx
n ! 1
1
n
xn
As n → ∞, the factorial growth is larger in the bottom than the exp. growth in the top.
. allfor 0! 1
as Therefore,1
xn
xn
n
This means that Rn(x) → 0 for all x.
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Remainder Estimation TheoremIf there are positive constants M and r such that
. ! 1
11
n
axrMxR
nn
n
11 nn Mrtf
for all t between a and x, then the remainder Rn(x) in Taylor’s Theorem
satisfies the inequality
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).
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Example Proving Convergence
3. Use the Remainder Estimation Theorem to prove the following for all real x.
0 ! k
kx
k
xe
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 .arbitrary and 0between for by bounded is 11 xtMretf ntn
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.
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Euler’s Formula
xixeix sincos
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Quick Quiz Sections 9.1-9.3
20
2
2
2
1. Which of the following is the sum of the series ?
(A) -
(B) -
(C) -e
(D) -
(E) The series diverges
n
nn ee
e
e
e
e
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Quick Quiz Sections 9.1-9.3
20
2
2
2
1. Which of the following is the sum of the series ?
(
(
A) -
(B) -
(C) -e
(E) The series diverges
D) -
n
nn e
e
e
e
e
e
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Quick Quiz Sections 9.1-9.3
2
2. Assume that has derivatives of all orders for all real numbers ,
(0) 2, '(0) -1, ''(0) 6, and '''(0) 12. Which of the following
is the third order Taylor polynomial for at 0?
(A) 2 3
f x
f f f f
f x
x x
3
2 3
2 3
2 3
2
2
(B) 2 6 12
1(C) 2 3 2
2(D) 2 3 2
(E) 2 6
x
x x x
x x x
x x x
x x
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Quick Quiz Sections 9.1-9.3
2
2. Assume that has derivatives of all orders for all real numbers ,
(0) 2, '(0) -1, ''(0) 6, and '''(0) 12. Which of the following
is the third order Taylor polynomial for at 0?
( ) 2A 3
f x
f f f f
f
x
x
x
2 3
2 3
2
3
3
2
(B) 2 6 12
1(C) 2 3 2
2(D) 2 3 2
(E) 2
2
6
x x x
x x x
x x x
x
x
x
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Quick Quiz Sections 9.1-9.3
0
0
0
0
0
3. Which of the following is the Taylor series generated by
( ) 1/ at 1?
(A) 1
(B) 1
(C) 1 1
1(D) 1
!
(E) 1 1
n
n
n n
n
n n
n
n
n
n
n n
n
f x x x
x
x
x
x
n
x
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Quick Quiz Sections 9.1-9.3
0
0
0
0
0
3. Which of the following is the Taylor series generated by
( ) 1/ at 1?
(A) 1
(B) 1
(C) 1 1
1(D) 1
!
(E) 1 1
n
n
n n
n
n n
n
n
n
n
n n
n
f x x x
x
x
x
x
n
x
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Pg. 386, 7.1 #1-25 odd