- Homework for 8.1 returned. Comments from the grader:
- #13, 14: Several people, when using Simpson's rule, would
take
and
and say
- #24: Most don't know what to choose for the height or the radius.
- #28: Mixed results, some just didn't know what they were supposed to do with errors
- #54: Most got
wrong.
- #56: Most couldn't figure this out.
- #13, 14: Several people, when using Simpson's rule, would
take
- Homework for 8.6 due today.
- Taylor Homework due Thursday (and I'm going to hold you to that!)
- Taylor Quiz returned
- Taylor polynomials are used for approximation, but approximating a function (not area! -- that was the job of our integration techniques).
- Taylor polynomials generalize linearization; they go beyond linearization.
- The linearization is
, the linear Taylor polynomial at the point of interest.
- Read Section 11.1 for next time, responding to the worksheet for section 11.1.
- The big picture: we're approximating functions (some nasty) with nice ones (polynomials):
- Check out these animations
- Taylor Polynomials Lab (from p. 213 of Stewart's Calculus)
- The Taylor Polynomials approximate functions:
But, in approximating, we make errors: let's talk error now. Remember the error terms in the integration schemes (which approximate areas)? As we discussed for the case of Simpson's rule, it's nice to give an estimate -- but even nicer when you can provide an error estimate, also. Well, turns out that we can in the case of Taylor polynomials, too.
- Examples:
- #23, p. 500
- How do the Taylor and Maclaurin polynomials compare?
- animation
- #35, p. 500 (sin(x) as well -- Maclaurin polynomial)
- #46, p. 501
- #23, p. 500
