1 / 10

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

sani
Download Presentation

Accuracy in Series

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related