What is the point of computations?

Computations get a bad name, as they are used in calculus class and
before as "busy-work" to keep idle students busy.
(and don't feel all happy-fuzzy and `creative')

Students "buy into" this: by the time they get to college, many think
math class consists of nothing more than applying an algorithm or
formula repeatedly: some find this boring, others get a sick pleasure
out of it.

Computations are useful, but should be -marked- as "just
computations": something to learn by rote, with a bit of understanding
on the way.

The idea is that once you can do computations, -then- you can get
understanding: if you can't do computations, you're unlikely to
understand what's happening, so much is your brain distracted by
trying to churn through the computations.

Computations can be a -road- to understanding, but should -always- be
marked as a -step-.

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What I mean is:
when doing something like "integration by parts",
show people -how- to do it.
Make it -blindingly- clear how to do the algorithm;
note exactly the steps where you might need to be careful.
Note that the -act- of -you-, the -teacher-, making the algorithm
clear will make you appreciate how tricky and subtle (and frequently
-stupid-) it is: frequently teachers who've learned how to apply the
machine fail to appreciate how tricky it is.

Also point out tricks/techniques (eg., how to recognize what to do in
an integration by parts/u-substitution) -- the goal is to get the
students -doing- the computation.

At first, understanding is -useless- and frankly a -hinderance-:
if you try to learn to compute -and- understand at the same time,
you'll have trouble with both. Learn the easy -computation- first!

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This is a -key- problem with high school calc books:
they -mainly- teach doing stupid computations
-but- they -bulk up- the presentation by including countless proofs
(which are ignored) and "understanding" which is likewise ignored.

Make the following separate:
- computation HOW-TO
- computation -understanding-
- application

EG: integration by parts:
(1) HOW-TO
given $\int xe^x$, write:
$u=x, dv=e^x dx$
then write:
$ u=x,  dv=e^x dx$
$du=dx,  v=e^x$
(NB: note that I write these -separately-, instead of on top of each
other: write things separately to -show- how an algorithm is done
-over time-)
(NB: note the care we take to "balance the $dx$": this is a good
mneumonic to make sure you're doing the computation correctly)
now do $uv - \int v\du$
(NB: I draw circles/ovals/lines on the picture to show what you
multiply: note again that the $dx$ help you know what to put inside
the integral)
...then you can finish to
$xe^x - e^x$

hint: let $u$ be something that gets easier on differentiating, and
$dv$ be something you -can integrate-
(eg., don't let $dv$ be $\arctan$!)
(specifically: inverse trig, $\log$, poly, exp (in that order of
 priority) work well for setting $u$)

trick 1: set $dv=dx$ (eg., $\int \log x$)

trick 2: get a recurrence (eg., $\int x \sin x$)

(2) understanding
this is the product rule/Leibniz rule backwards:
$(uv)'=u dv + v du$
integrating yields:
$uv=\int u dv + \int v du$
rearranging yields:
$\int u dv = uv- \int v du$

(3) application
fourier series, say

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see how big the algorithm is?
we show how it's done in detail,
and flag tips and tricks

the actual "take-away" understanding is easy by comparison,
and applications are rapid.

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Algorithms/techniques students learn in calc class:
(make a hand-out: here are the techniques you'll use/learn)
(also a good "take home": the techniques you learned in calc class)

A: derivatives
- specific rules:
 $x^n, \log x, e^x, \log_b, b^x$, trig, inverse trig (urk!)
- general rules:
 linearity, product rule, quotient rule (urk), chain rule
- maxmization: first/second derivative test
  (second derivative test is useful for Taylor series)
- graph sketching
  long, intricate: separate handout

B: integrals
- backwards specific differentiation rules
- backwards general rules:
 linearity, by parts, $u$-sub
- rational functions:
  + polynomial long division
  + partial fraction decomposition
    (over $\bC$, please: else get ugliness)
  + integration of partial fractions
- stupid integral tricks:
  + trig substitutions
- solving ODEs:
  + linear homogeneous
  + separable
  + (urk: $p(x)y'=q(x)$?)

C: power series
- convergence tests
- taylor series/polynomials
- remainder term (urk)
- differentiating/integrating power series termwise

X: limits
- plug-in if continuous
- rational functions:
  + cancel factors
  + leading term
- L'H\^opital's rule
