Calculus II - 30
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Calculus II - 30

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Stewart Calculus 11.11

Stewart Calculus 11.11

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Calculus II - 30 Calculus II - 30 Presentation Transcript

  • 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 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 ?
  • What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?
  • What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | !
  • What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | ! ( ) | |=|− |≤ , =
  • What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | ! ( ) | |=|− |≤ , =when − . ≤ ≤ .
  • What is the maximum error possible in usingthe approximation ≈ − ! + !when − . ≤ ≤ . ?Taylor’s Inequality: | ( )| | | ! ( ) | |=|− |≤ , =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 than ?
  • What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! !
  • What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! ! . | ( )| | | . ! !
  • What is the smallest degree of the Taylorpolynomial we can use to approximate if we wantthe error in [ . , . ] to be less than ? . | ( )| | | . ! ! . | ( )| | | . ! ! . | ( )| | | . ! !
  • 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.
  • 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 . + = ( ) = + + ··· = ( + )! ! ! !