Day 7, Mat128: Calculus A

• Reminder: turn off your rectangles.
• Your quizzes are returned, as is your first homework.
  • quizzes: you're getting the definition of function down, at any rate! Some of you still don't know, however....
  • homework: randomly chose 2 and 7 to grade.
    • Each is graded out of two, so your assignment is out of 4.

    • Answer each question completely. On problem 2, for example, it asked "What are your conclusions?"

    • Don't forget units. Units can be very helpful. For example, in problem 2 you're looking for heart rate (beats per minute). This suggests the ratio (difference in heart beats)/(elapsed time). Some of you had that reversed.

    • You need to show work, especially on the odd problems where the answers are printed in the back....

• You have an assignment for section 1.5 due Wednesday.

• #18, p. 60
• #26, p. 60
• #35, p. 61
• #42, p. 61

Questions about anything at all (including homework for 1.5), before we move along?

• Limit laws:
  • We start with two especially trivial special limits:
    • Constant rule:

    • Identity rule:

    What can we do with those? We need more horsepower....

  • Sum law: "the limit of a sum is the sum of the limits"

    So what is this limit:

    

    This sum (and many other operations) satisfy the same pattern:

    "The limit of a STUFF is the STUFF of the limits."

    So, in particular,

    "The limit of a sum is the sum of the limits."
    

  • Difference law
  • Product law
  • Quotient law (slight caveat here -- no zero denominators)

  • Power law
  • Root law -- again, a caveat: why can't we compute

    \[ \lim_{x\to{0}}\sqrt{x} \]

    Because the function isn't even defined as x approaches 0 from the left.

  • Putting these laws together
    • Now that we've got the sum and product and constant multiple rules, what can we deduce?
    • "Affine" rule, for one:
      \[ \lim_{x\to{a}}(mx+b)=ma+b \]

    • Then we can proceed to deduce that all polynomials have limits, and furthermore that
      

      This is the conclusion of Exercise #55, p. 71, which says that polynomials are continuous, which we'll consider more in section 1.8.

      Exercise #56 says it's also true for rational functions, for every element of their domain.

  • Other Laws
    • The squeeze theorem What can we say about the limit of g as x approaches a?

  • Some Examples

  • Additional examples:
    • #63, p. 71
    • #59, p. 71 (squeeze it!)