Day 40 in MAT229

Today:

• Announcements:
  • Return Homework 11.5: graded #38, 45

  • Homework 11.6 due

  • Tomorrow your special 11.6 assignment is due

• 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 n 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:

    

  • Examples:
    • #1, p. 596
    • #4, p. 596 (remember that there are several different ways to do these! You don't have to work from the definition....)
    • You choose some....