1 / 12

11.6 Taylor’s Theorem: Error Analysis for Series

11.6 Taylor’s Theorem: Error Analysis for Series. Tacoma Narrows Bridge: November 7, 1940 . Greg Kelly, Hanford High School, Richland, Washington. Graphically estimate the values of x for which the Taylor polynomial approximates

feng
Download Presentation

11.6 Taylor’s Theorem: Error Analysis for 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. 11.6 Taylor’s Theorem: Error Analysis for Series Tacoma Narrows Bridge: November 7, 1940 Greg Kelly, Hanford High School, Richland, Washington

  2. Graphically estimate the values of x for which the Taylor polynomial approximates to four decimal places.

  3. Taylor series are used to estimate the value of functions (at least theoretically – now days we can usually use the calculator or computer to calculate directly.) An estimate is only useful if we have an idea of how accurate the estimate is. When we use part of a Taylor series to estimate the value of a function, the end of the series that we do not use is called the remainder. If we know the size of the remainder, then we know how close our estimate is.

  4. Use to approximate over . ex. 2: Since the truncated part of the series is: , the truncation error is , which is . For a geometric series, this is easy: When you “truncate” a number, you drop off the end. Of course this is also trivial, because we have a formula that allows us to calculate the sum of a geometric series directly.

  5. Lagrange Form of the Remainder Taylor’s Theorem with Remainder If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I: Remainder after partial sum Sn where c is between a and x.

  6. Note that this looks just like the next term in the series, but “a” has been replaced by the number “c” in . Lagrange Form of the Remainder This seems kind of vague, since we don’t know the value of c, but we can sometimes find a maximum value for . Remainder after partial sum Sn where c is between a and x. This is also called the remainder of order n or the error term.

  7. If M is the maximum value of on the interval between a and x, then: Taylor’s Inequality Lagrange Form of the Remainder

  8. error Find the Lagrange Error Bound when is used to approximate and Error is less than error bound. Lagrange Error Bound

  9. Find the Lagrange error bound that results when is replaced with Support graphically.

  10. The first three derivatives of are: • Find the 2nd degree Taylor polynomial about x=4. • Use your answer to estimate the value of f(4.1). • Find a bound on the error for the approximation.

  11. The first three derivatives of are: • Find the 2nd degree Taylor polynomial about x=4. • Use your answer to estimate the value of f(4.1). • Find a bound on the error for the approximation.

  12. Homework Page 519 #1-9,15-18

More Related