Day 12 in Mat129

Today:

• Announcements
  • We have an exam one week from today.
  • Hand in pp. 106--, #69-72
  • New assignment:

    Mon

    2/8

    Section 3.2-3.4

    Begin on-line homework assignments 3.4 (due Tuesday, 2/16) By hand: pp. 137, #52-54 (due Thursday, 2/18)

• Roll

• Section 3.2: The Derivative Function:
  • The definition:
    

  • Here's an example of a little trickier function: #6, p. 116

  • Derivatives of
    • constant functions
    • the identity function f(x)=x

  • Rules that help us avoid the definition! (with Proofs)

    • The sum rule

      works as we'd hope: the derivative of a sum is the sum of the derivatives

    • The constant multiple rule (makes sense)

    • The power rule (via dominoes -- i.e., mathematical induction)

    • NOTE: at this point we have all the rules necessary to differentiate all polynomials, without needing to resort to the definition!

  • #63 and 64, p. 118

  • Note Theorem 3 (p. 113): differentiability implies continuity (we know that already!)

  • But continuity does not imply differentiability -- examples?

• Section 3.3

  • The product rule

    Rats! "The derivative of the product is not the product of the derivatives."

  • Now we can get all polynomial derivatives.

  • The quotient rule (Rats again!). We can get this one in two steps:
    • we calculate
    • and the product rule.
  • Now we can get all rational function derivatives!

  • Examples:
    • #2, p. 124
    • #8, p. 124
    • #27, p. 125
    • #38, p. 125
    • #42, p. 124

• Section 3.4: Rates of change
  • We know of the derivative as related to rates of change.

  • Examples:
    • Rate of change of area of a circle with respect to radius
    • Linear Motion
    • Motion under the influence -- of gravity!
    • #30, p. 135
    • #32, p. 135
    • #46, p. 136

Links:

• Javamath