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Accuracy in Series

In Lesson 9.7B, we explore the accuracy of Taylor polynomial approximations through the concept of the remainder. We discuss the actual and approximate remainder functions and the error associated with these approximations. The lesson covers how to determine maximum error using formulas based on the n+1st derivative and good decimal digits. We illustrate techniques for both odd and even term series and provide shortcuts for common functions. Exercises include calculating the fifth Maclaurin polynomial for sin(x) and determining the degree needed for ln(1.3) with a specified error.

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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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