(Should put these in the sections, with appropriate CAUTION macros,
 so that they are typeset correctly -and- can be extracted automagically.)

general mistakes/techniques:

- visualize functions
  be able to sketch, roughly:
  $x^2, x^3, \sqrt{x}, x^{1/3}, e^x, e^{-x}, \ln, \cos, \sin, \tan$
- check sign
  if $f$ is increasing, then $f'\geq 0$;
  if $f$ is non-negative, then $\int_a^b f$ is non-negative (often
  positive)
- estimate/orders of magnitude
  $19^3 < 8000$, because $19 < 20$, but it's not much less
  $xe^{-x} \to 0$ as $x \to \infty$ b/c exponential growth dominates linear
  $e \approx \pi \approx 3$, etc.
- variables are better than numbers
  if you have variables, you can check if a formula makes sense, then
  plug in the numbers; if you just have numbers, you can't check the
  -formula-
- symmetry
  if variables appear in a symmetric form in one form of an
  expression, they should appear symmetrically in other forms
  (e.g., the area of an ellipse can't possibly be $\pi a^2 b$, as $a$
  and $b$ should be symmetric)
- special cases/extreme cases
  if finding formula for area of ellipse, make sure it works for
  circle, and in case one axis goes to $0$ or $\infty$
- dimensional analysis
  note that you can't plug dimensioned numbers into analytic functions
  (as the dimensions will go off to infinity) or transcendental functions
  in general; they only make sense in rational algebraic expressions
- arithmetic errors
  adding when you should subtract, multiplying when you should divide,
  especially dealing with fractions, negative powers, etc.

-------------------
- thinking 0/0 is undefined as a limit
  (NB: in the derivative, the limit is -always- 0/0!!! -- even though
   the limit is an actual number!)

- being careless with infinities
  (e.g., infty/infty = 1...) (people should -not- say: that's
  $\infty^2/\infty = \infty$: that's good intuition, but you need to
  justify/validate it. e.g., what's $e^\infty$? you need to make sense
  of this)
-------------------
- increasing implies $f'>0$ (not!)
  (already filed)
- u-substitutions in definite integrals:
  forgetting to either:
   (a) back substitute for $u$
   (b) change limits of integration
  (great example: volume of sphere by shells: if they mess up
    $u=1-x^2$, they'll be missing a $-1$, and they'll get a negative volume)
- shells: if you have a vertically symmetric shape, forgetting that you need to
  integrate $2f = f - (-f)$, not just $f$.
- absolute value mistakes:
  confusing $\int \abs{f}$ and $\abs{\int f}$
    in case of paths, first is total distance traveled,
    and second is magnitude of displacement.
