Day 21 in Mat228

Today:

• Announcements
  • Homework for 11.6 postponed until Tuesday, 11/10.
  • Exam revisions graded: come in to discuss it and get 100% on the redo. Email me to schedule a meeting (Please use subject line REDO).

    Only 6 out of 14 have contacted me about this! What are the rest of you waiting for?

  • Quiz for 11.7 returned.
  • Next time we'll be starting "something completely different": Chapter 13: vector geometry
  • Read Section 13.1 for next time responding to the worksheet for section 13.1.
  • What should we do about quizzes? They're not working, for you or for me!

• Any questions on old stuff?

• Section 11.7: Taylor Series
  • We've basically seen all this material before: we're now just summarizing:
    
    is the Taylor Series expansion of f(x) centered at x=c. The special case of c=0 is called the Maclaurin series. The factorials in the denominator sweep up all the powers that pop down during the differentiations.

    The coefficients a_kare "designed" so that after k differentiations of the series, when we evaluate the series we pick off only the k^th derivative evaluated at x=c:

    

    

    We've seen that, because of the radius of convergence, the function may not be represented by a power series at some values of x (even as it's perfectly represented at other values of x).

    If we can bound all derivatives of a function on an interval about x=c, then the Taylor series represents the function there:

    

    The expansion for the binomial series is a classic: it's another of the great achievements of Sir Isaac Newton, one of the founders of calculus:

    

    What about its radius of convergence? Let's look at #84, p. 599.

  • Examples:
    • #1, p. 596
    • #4, p. 597 (remember that there are several different ways to do these! You don't have to work from the definition....)
    • #10 p. 597
    • Let's take a look at some more interesting problems....
      • #48, p. 597
      • #49, p. 598
      • #77, p. 598 (homework)
      • #79, p. 598