Day 16 in Mat121

Today:

• Note: quiz Thursday over definition of the derivative

• directed reading handout, Section 3.5 (html) (pdf)

• Sections 3.2 and 3.3: The Derivative Function, and important rules
  • Rules that help us avoid the definition! (with Proofs)
    • The power rule (via dominoes -- i.e., mathematical induction)
      1. Prove that P(1) is true.
      2. Show that if P(k) is true, then P(k+1) is also true.
      Then P is true for all integers.

    • The product rule

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

    • 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 -- it follows from the product rule)

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

    • The quotient rule -- rats again! Watch the sign:
      Low d High minus High d Low
      Over the denominator square they go!

  • Differentiability implies continuity

  • Examples:
    • #61, p. 118
    • #63, p. 118
    • #64, p. 118