Upcoming SlideShare
×

# Calculus II - 30

407 views
360 views

Published on

Stewart Calculus 11.11

Published in: Technology, Education
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
407
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
16
0
Likes
0
Embeds 0
No embeds

No notes for slide
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• \n
• ### Calculus II - 30

1. 1. 11.11 Applications of Taylor Series −Evaluate correct to within an errorof 0.01.
2. 2. 11.11 Applications of Taylor Series −Evaluate correct to within an errorof 0.01. − ∞ (− ) ∞ = = ! = = (− ) ! = − ! + ! − ! + ···
3. 3. 11.11 Applications of Taylor Series −Evaluate correct to within an errorof 0.01. − ∞ (− ) ∞ = = ! = = (− ) ! = − ! + ! − ! + ··· − = − · ! + · ! − · ! + ··· +
4. 4. − ∞ (− ) ∞ = = ! = = (− ) ! = − ! + ! − ! + ··· − = − · ! + · ! − · ! + ··· +
5. 5. − ∞ (− ) ∞ = = ! = = (− ) ! = − ! + ! − ! + ··· − = − · ! + · ! − · ! + ··· + − = − · ! + · ! − · ! + ···
6. 6. − ∞ (− ) ∞ = = ! = = (− ) ! = − ! + ! − ! + ··· − = − · ! + · ! − · ! + ··· + − = − · ! + · ! − · ! + ··· = − + − + − ···
7. 7. − ∞ (− ) ∞ = = ! = = (− ) ! = − ! + ! − ! + ··· − = − · ! + · ! − · ! + ··· + − = − · ! + · ! − · ! + ··· = − + − + − ··· ≈ .
8. 8. What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?For what value of is this approximationaccurate to within . ?What is the smallest degree of the Taylorpolynomial we can use to approximate if wewant the error in [ . , . ] to be lessthan ?
9. 9. What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?
10. 10. What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | !
11. 11. What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | ! ( ) | |=|− |≤ , =
12. 12. What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | ! ( ) | |=|− |≤ , =when − . ≤ ≤ .
13. 13. What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | ! ( ) | |=|− |≤ , =when − . ≤ ≤ . . . !
14. 14. For what value of is this approximationaccurate to within . ?
15. 15. For what value of is this approximationaccurate to within . ?We want | | < . !
16. 16. For what value of is this approximationaccurate to within . ?We want | | < . ! | | < . · ! .
17. 17. For what value of is this approximationaccurate to within . ?We want | | < . ! | | < . · ! . | |< .
18. 18. What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ?
19. 19. What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! !
20. 20. What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! ! . | ( )| | | . ! !
21. 21. What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! ! . | ( )| | | . ! ! . | ( )| | | . ! !
22. 22. What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! ! . | ( )| | | . ! ! . | ( )| | | . ! !So we need the Taylor polynomial of degree 8.
23. 23. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
24. 24. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
25. 25. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
26. 26. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
27. 27. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
28. 28. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
29. 29. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
30. 30. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
31. 31. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
32. 32. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
33. 33. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
34. 34. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !
35. 35. Taylor approximation of . + = ( ) = + + ··· = ( + )! ! ! !