Day 5 in Mat129

Today:

• Announcements
  • Hand in written problems
  • Results of KEMTP test
  • New assignment:
    • Read 2.4.
    • Begin problem set 2.3 on IMathAS (due Thursday, 1/28, at midnight).

• Roll

• Section 2.2: Limits: A numerical and graphical approach
  • Definition of a (finite) limit
    • What is it about real data that causes trouble?
    • We want to look at functions defined by formulas, so that the idea of a limit makes sense. For that reason, we leave behind the Keeling curve for the moment.
    • So we've got to be able to refine our subdivisions continually, forever, more finely....
    • Consider f(x)=|x|
    • Theorem 1
      • Proof of the constant rule
      • Proof of the linear rule
      These will be generalized in the next section.
  • Graphical and Numerical Investigation
    • Let's take a look at #2, p. 57, and perform an investigation
    • This is an example of a function with a hole in it.
    • What kind of functions have "holes in them"?
    • What else can go wrong?
  • One-sided limits
    • What functions would be different on one side of a point than they are on the other?
      • postage
      • general costs of items in bulk
  • Infinite limits
    • There are two kinds of infinite limits.
    • In this section, we're horning in on a constant value of x, and finding that the function is growing "out of bounds"
    • Example:

• Section 2.3: Basic Limit Laws
  • Sum law
  • Constant multiple law
  • Product law
  • Quotient law

  • Examples:
    • p. 62, #8
    • p. 62, #10
    • p. 63, #23
    • p. 63, #31