- Your first test is returned.
- Curve: add 5 to your score
- The Exam
- A key
- Comments:
- We've got to know definitions (e.g. rational functions).
- We've got to know properties (e.g. continuity).
- We need to know the law(s) (be able to justify our actions)!
- Many of you don't know the sine function. It's one
of the most important functions known to us.
It's periodic -- it repeats -- and it's a beautiful wave, as so many things in nature are.
- Some of you may want to make an appointment to see me.
- You have a homework due. Questions?
We'll review, then use the rules obtained to find derivatives of all polynomials, then prove the quotient rule.
These rules free us from an incredible amount of algebra, so are much beloved by calculus students. They fall in love with the rules, however, rather than with the definition that gave rise to them. Better to love the definition: the really powerful idea is a limit,
- Rules that help us avoid having to use the definition each time (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 product rule (has a tricky bit)
- 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!
The derivative of the monomial $ax^n$ is $nax^{n-1}$.
And a polynomial is just a sum of these. So
$s'(t)=(at^2+bt+c)'=2at+b$
and
$s''(t)=(2at+b)'=2a$
-
- Other rules:
- The quotient rule:
``Lo dee hi minus hi dee lo, over the denominator square we go!''
- Personally I prefer to turn quotients into products, of
the form \(\frac{f(x)}{g(x)}=f(x)\frac{1}{g(x)}\), then use the
product rule.
- The quotient rule:
- Examples from 2.3:
- #13, p. 136
- #22, p. 136
- #48, p. 137
- #56, p. 137
- #58 (we'll need the quotient rule)
- #64