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Accuracy in Series

Accuracy in Series. Lesson 9.7B. Remainder of a Taylor Polynomial. We need a sense of how accurate our approximation is Actual Approximate Remainder Function Value. Remainder of a Taylor Polynomial. Error associated with the approximation

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Accuracy in Series

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  1. Accuracy in Series Lesson 9.7B

  2. Remainder of a Taylor Polynomial • We need a sense of how accurate our approximation is • Actual Approximate RemainderFunction Value

  3. Remainder of a Taylor Polynomial • Error associated with the approximation • We can determine the maximum error with the formula

  4. Error Calculation Where … • M is the bound on the n+1st derivative of f(x) • d is the number of good digits after the decimal

  5. Error Calculation • When series is all odd terms • Replace (n + 1) with (2n + 3) • When series is all even terms • Replace (n + 1) with (2n + 2)

  6. Error Calculation • We will be given • f(x) … from this we can determine M • c … the center • Thus, given any two of x, n, and d you can determine the other one

  7. Error Calculation • Given f(x) we determine M for the interval [a, b] spanned by c and x • Shortcuts • If f(x) = sin(x) or cos(x), then M = 1 • If f(x) = ex then • If f(x) = e-x then • Note signifies the "ceiling" function, the next integer beyond the largest value in the interval • Note remaining shortcuts on handout

  8. Try It Out • Fifth Maclaurin polynomial for sin x • Determine P3(0.2) • Use to determine the accuracy of the approximation

  9. Try It Out Some More • Determine the degree of the Taylor Polynomial Pn(x) expanded about c = 1 that should be used to approximate ln(1.3) so that the error is less than 0.0001 • We are given • d • x • We seek the value of n • Note the interval is [1, 1.3]

  10. Assignment • Lesson 9.7B • Page 659 • Exercises 45 – 59 odd

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