Day 9 in Mat129

Today:

• Announcements
  • Return of problems pp. 59--, #53, 56, 61
    • First of all, fewer than half handed it in....
    • "Numerical Evidence" -- you were really supposed to dig out the ol' calculator and create a table or two, to indicate what's going on....
    • #56 involves a conjecture, which we can now prove!
    • #61 sets us up for today's introduction of the derivative
  • You have a problem set due tomorrow (by hand).
  • New assignment:

    Mon

    2/1

    Section 2.7/3.1

    Read 3.2 in preparation for class tomorrow Hand in the following for grading: pp. 106--, #69-72 (due Monday, 2/8)

• Roll

• 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

• Section 3.1: The Definition of the Derivative
  • "time, position, velocity"
    • If we have the record of the position, could we recover the velocity?
    • If we have the record of the velocity, could we recover the position?

  • Secant lines approximating tangent lines....

  • The definition:

  • Example problems:
    • #5, p. 103
    • #21, p. 104
    • #26, p. 104
    • #49, p. 105

• Javamath