Day 29 in Mat222

Today:

• Collect problems 12.12.

• Questions/Comments on old stuff?
  • Problems:
    • (p. 807, section 12.12) #35: Newton's method and Taylor's inequality
• Review:
  • The Final:
    • Chapter 12, sections 8-12, will be featured as half of the final;
    • the other half will be old material: Chapter 8, sections 1, 4, 7, 8; and Chapter 12, sections 1-7 (section 7 provides a good review of results through sections 1-6).
    • I've reserved the room from 4:30 on the day of the final, and I'll be here to answer last minute questions.

  • Sections 12.8-12 feature several concepts:
    • power series:
      • It all started with 1/(1-x): we found that this is equal to 1+x+x^2+x^3+...., as long as x is smaller than 1 in size.
      • Idea: we can approximate arbitrarily well a function by an "infinite polynomial" (on some interval). The approximation is great at a point (the center of the series), and generally falls off as we get farther away from the center.
      • center of the series (usually denoted a)
      • radius of convergence (the series is only equal within this radius, and possibly at the endpoints).
      • interval of convergence (checking endpoints!)
      • creating new power series by
        • composition
        • integration
        • differentiation
      • Using alternating series to bound error

    • Taylor series
      • Maclauren series (centered on 0)
      • Coefficients given by derivatives of f
      • Coefficients chosen so as to match f at a in terms of higher and higher derivatives.
      • Taylor polynomials provide approximation to the original function.
      • Error in using finite polynomial bounded by Taylor's inequality (useful if we can bound the n+1^thderivative)
      • Plotting Taylor polynomials using your TI (using the taylor command)
        • evaluating "fit" to polynomial approximation
        • evaluating error on given interval

  • True-False quiz, p. 810

  • Problem 58, p. 811

Next time: Further Review and Evaluations