Some cute maximization problems:
1 -----
 a: if f is increasing,
   then fg is maximized iff g is maximized
   (fg reaches its max when g reaches its max)
 
 b: using the above, find the max of
   (hideous increasing function, applied to a nice g,
    like
    $[\tan (x^{3/17})^{1/3}]$
    applied to something inside)
------------------------------------------
Here's another one, to do -early-
[IE, -before- you get to Taylor series]:
[mention that this will be useful later,
 but it's also a way of codifying patterns they see]
write down the general form of the derivatives of stuff like:
$x^{-n}$, $\sqrt{x}$

The merciful thing to do is define for 'em:
[the ``falling factorial'']
$(n)_k = n(n-1)\dots(n-k+1)$
and have 'em show that the $k$th derivative of $x^n$
is $(n)_k x^{n-k}$.
[the recurence they need to use (and obviously show)
 is $(n)_k (n-k)=(n)_{k+1}$, which is immediate from writing it out]

[this is -much- cleaner than most other ways they'll have to do it]
[and then ask them to write out some of 'em:
 - general form for $x^{-n}$, using factorials
 - general form of a derivative of $\sqrt{x}$, by hand,
   using double factorials ($n!!$) and $2^n$.]
Note that this is a good EG of how good notation makes life easy.
...and this is a good notation: it's $nPk$!

Have 'em give an informal proof
(duh, this is the pattern) -- which was fine in Bernoulli's day,
and then give a formal inductive proof:
- it's true for $k=0$
  (or maybe $k=1$: make sure they're careful about this!)
- the derivative of the form they claim for $k$
  gives the form for $k+1$.
  [this is very easy with the falling factorial notation]
