11.11 Applications of    Taylor Series           −Evaluate       correct to within an errorof 0.01.
11.11 Applications of    Taylor Series                −Evaluate                    correct to within an errorof 0.01. −   ...
11.11 Applications of    Taylor Series                  −Evaluate                        correct to within an errorof 0.01...
−           ∞        (− )               ∞        =    =          !      =         =   (− )       !        =   −    !   +  ...
−           ∞        (− )                 ∞        =    =          !        =         =   (− )         !        =   −    !...
−           ∞        (− )                 ∞        =    =          !        =         =   (− )         !        =   −    !...
−           ∞        (− )                 ∞        =    =          !        =         =   (− )         !        =   −    !...
What is the maximum error possible in usingthe approximation                 ≈ −      !   +   !when   − . ≤   ≤ .    ?For ...
What is the maximum error possible in usingthe approximation                ≈ −      !   +   !when   − . ≤   ≤ .   ?
What is the maximum error possible in usingthe approximation                  ≈ −       !   +     !when   − . ≤     ≤ .   ...
What is the maximum error possible in usingthe approximation                        ≈ −      !   +     !when       − . ≤  ...
What is the maximum error possible in usingthe approximation                        ≈ −      !   +     !when       − . ≤  ...
What is the maximum error possible in usingthe approximation                        ≈ −             !   +     !when       ...
For what value of        is this approximationaccurate to within   .           ?
For what value of        is this approximationaccurate to within   .           ?We want             | |                  <...
For what value of        is this approximationaccurate to within   .           ?We want             | |                  <...
For what value of        is this approximationaccurate to within   .           ?We want             | |                  <...
What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less ...
What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less ...
What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less ...
What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less ...
What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less ...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
Taylor approximation of             .                         +     =       (   )              =           +           + ·...
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Calculus II - 30

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Stewart Calculus 11.11

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  • 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 . + = ( ) = + + ··· = ( + )! ! ! !

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