Day 03 in Mat122

Today:

• Directed reading handout: Section 5.2 (html) (pdf)

• Summary of Newton's method, from yesterday:
  • Objective: finding roots; that is, values of x such that f(x)=0.
  • Strategy:
    • given a good guess, x=a,
    • find the linearization about (a,f(a)), L(x)=f(a)+f'(a)(x-a) and
    • solve L(x)=0 to find the iterative formula x=a-f(a)/f'(a)
  • Does it work? With three "iterates", we computed the square root of 7 to 8 decimal accuracy!
  • Your calculator uses it...

• Section 5.1: approximating and computing area
  • We'd made a start on solving the following problem:

    Suppose that you clock your speed at each minute along a trip, in mph:

    minute	0	1	2	3	4	5	6	7	8	9	10
    speed (mph)	0	30	45	30	70	65	70	70	45	30	35

    Estimate the distance travelled.

  • Summation notation

  • Bogdan's rule: d=44.54545454545455*10 = 445 miles*minutes/hour = 7.42 miles

  • Trapezoidal rule
    • Assumes continuity of underlying speed function
    • We fit straight line segments between data points

  • Left/Right/Midpoint rules
    • relax continuity! Use rectangles...
    • also called rectangle rules: lrr, rrr

  • Basically, every function can be used to create a table like this, and then we can use these strategies.

  • Here's an important theorem:

  • Examples:
    • #5, p. 258 (also trapezoidal)
    • #9, p. 258