lecture:
  example in 2D (sin theta, r sin theta)
functions as mappings: ask 'em to graph tan(x^3) (and tan^3(x))

write up worksheet on f (number line), f' (number line), f'' (number line)
  right now: graphing.text

chapt 4
  review: meaning of sign of 1st/2nd der; max/min (1st, 2nd der. test)
	functions as mappings (my thing: e.g., tan(x^3))
	curve sketching

-- brief note on exp & log --
my defn: e ~= 2.7 (does kathleen know more digits? ;-)
  (~2.71828182845904523536)
	10^{\sqrt{2}} should be limit as x=p/q \to \sqrt{2} of 10^x
	(does make sense, as 10^x is continuous and increasing on Q,
	 but requires more sophistication than we have)
   so e^x ~= 2.7^x
	more properly, e = 1 + 1 + 1/2! + \dotsb
	and obviously converges to a number a little less than 3 but more than 2.5
	(geometric series comparison)
	log is the inverse
	derivative of e^x, log x
	  (using 2^x as justification; use chain rule to compute other way)
	(no discussion of other bases -- they'll come later)

chapter 5
- volume of cone tricky: use ??'s principle and dilation/scaling, and cut apart
 a cube into 3 pieces (i.e., once we've computed it for the square cone,
  we can compute it for any cone) (note -- we're using 3-fold symmetry of cube! ;-)
- do archimedes' volume of sphere
historical note: Newton and company didn't feel a need to be rigorous about what
they did: of -course- they were computing area (whatever that means)
rigor came in the 19th century, and with it generalizations in the 20th century
(rigor is good! it makes you think proper-like)

  emphasize that we are only working with -bounded- functions on -compact- intervals
	- show why (by inhomogenous dilation) the integral of x^n should be what it is
	  (remark that there's a trick (in monthly) which shows the constant factor)
	- upper & lower sums: Darboux method (J.G.Darboux, 1842-1917)
	  ...this shows that we need sups and infs
	- Riemann sums (rough estimate -- actually works) (Riemann, 1826-66)
	  (FIXME: don't see why Riemann integrable implies Darboux integrable
		  (converse is obvious))
	- pf that riemann sums work for uniformly continuous functions
	  (unif. continuous is -almost- exactly what we need;
		 FIXME: what is exactly what we need? bounded variation?)
	- F(x) and anti-derivative
	  (notion of elementary function -- Newton's idea of a function:
		  something we can compute with)
	- Fundamental theorem!!! (super-cool)
	- integral is -signed- area (need this to make it linear)
	- techniques of integration:
	  + 5.6 = change of variables
		+ 8.2 = integration by parts
		(just do the derivative techniques backwards, since they don't work
		 properly "forward" -- not reversible)

  - 5.8: integral as average (applications to probability)
	     (center of mass too)
	  (mean is integral of x w/r/t measure P(x) where P(x) is prob. distro,
		 i.e., integral of x * P(x)dx -- you need to weigh certain points more)

chapter 6
  6.1 - 6.3
  when doing discs/shells, also do surface area (review arc length)
	also do n-dim'l volume:
after having done integrals: do multi-dimensional integrals (actually
only need to use 1 dimensional integration!) to compute volume of cone
(1/n volume of base, where n = # dims) -- yes, the 1/2, 1/3 pattern
of triangles and cones continues
volume of simplex = 1/n! (times k^n, where k is the scaling factor)
------------------------------------------------
		
chapter 7
  generalities about inverse functions
	  (linear algebra: (x,y) -> (y,x) is: linear; is reflection through y=x)
		inverse is diff/continuous if function is
		 FIXME: is this true at a point? (proof kinda tricky, non-obvious
		   -- won't prove; clear from picture, as continuous invertible functions are easy)
		derivative of inverse (formal exposition -sucks-: think of it as rate of
		change or (linear) stretching factor and then it's -obvious-)
  exp and log
	  via power series (treated informally)
		show via power series that exp(x)*exp(y) = exp(x+y)
		(homework: have students show that log(x*y)=log(x) + log(y)
		 !!! make sure this is doable, using power series log(1+x).
		 maybe just show log(x^2) = 2 log(x)?)
		via power series derivative is -easy-!
		different base (relate to standard exponent rules 2=e^log 2, so 2^x = e^log 2 ^ x)
  	growth! growth! (exponential vs. polynomial)
(the gudderamian (or whatever the fuck it is...)!)
  (radioactive decay is b/c each atom has independent odds of decaying)
(logarithmic differentiation)

etc.:
  pappus' theorems 6.4, 9.9 -- application of "integral is average"

linear algebra:
 - derivative is linear!
 - integral is linear (kinda sorta)!
 - that -damn- +C is the kernel of the derivative (as we showed earlier)

mathematical physics:
- differential equations: simple harmonic oscillators (and damping)
- work (6.5)
  ("integrals" of systems: conservation laws
	 e.g. of spring and potential/kinetic energy
	 conservative forces (i.e., closed forms) have (local) potentials,
	   i.e., are exact -- math!)
- fluid force/pressure (6.6)

chap 8 

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make database of exercises
fields: problem (as TeX? as MathML?)
  hint(s)
  note (discussion, importance, etc.)
        summary/verbal description (for browsing pleasure of instructor)
  keywords/topics (to help with queries)
  required knowledge/theorems
  difficulty??? (computation, application, insight?)
