Day 8 in Mat129

Today:

• Announcements
  • Hand in problems pp. 59--, #53, 56, 61 (due Thursday, 1/28, in class)
  • New assignment:

    Thr

    1/28

    Section 2.6/2.7

    Read 3.1. There are more chapter 2 practice problems on-line.

• Roll

• Section 2.5: Evaluating Limits Algebraically
  • What's the big idea? Indeterminate forms cause trouble.

    Strategy: transform f(x) algebraically if possible into a new expression that is defined and continuous at x=c and then evaluate by substitution.

  • Examples
    • #1, p. 77
    • #3, p. 77
    • #20, p. 78
    • #32, p 78

• Section 2.6: additional indeterminate limits
  • Important trigonometric limits:
    1. 
    2. 

  • These results can be proven with the so-called "squeeze theorem"
    1. The first limit is pinched between and

       (see graphical demonstration on p. 80)

    2. The second is pinched between and
    (for the second, see #32, p. 78)

• Section 2.4: Limits and Continuity
  • One more important theorem: Theorem 4, p. 68, RE composite functions
    • Roughly: A composition of continuous functions is continuous.
    • Let be a composite function. If is continous at and is continous at , then is continuous at .
  • Examples:
    • #20, p. 71
    • #34, p. 72
    • #45, p. 72

• Section 2.7: The Intermediate Value Theorem
  • A challenge....

  • The Intermediate Value Theorem:

    If f is a continuous function on the closed interval [a,b] and the number M satisfies f(a) < M < f(b) (note: the endpoints are not equal), there is a point c between a and b where f(c) = M.

  • What can go wrong if f(a)=f(b)?

  • One thing this gives us is a method for finding roots of functions to greater and greater precision -- The Bisection Method.

  • Examples:
    • Preliminary questions, p. 86: 1, 2, 4
    • #7, p. 86
    • #15, p. 86
    • #24, p. 87