bad freshman calculus

log_a(x) = ({{ln x}\over{ln a}}) = {1\over{x ln a}} by quotiant rule
EXHIBIT A

but this is *senior level* stuff…
“analysis” *not* freshman-calc.
and even *there*, one seldom encounters
such an *explicit* version of the
natural-log-equals-reciprocal fallacy
ln(x) = {1\over x}.
(the “quotiEnt rule”…
or rather, *the* quotient rule…
our subject has fallen into the
“leaving out articles makes it
harder to understand and so is
in my best interest” trap…
has *no bearing* on this [mis]-
calculation [we aren’t diff-
erentiating a fraction (or any-
thing else for that matter)].)

here’s the whole sad story.
math is hard and everybody knows it.
what they *don’t* know is that it’s
nonetheless easier than anything else.
*particularly* when one is trying
to do things like “pass math tests”.

there’s always a widespread (and *very*
persistent) belief abroad in math
classes that trying to understand
what the technical terms mean (for
example) is “confusing” and should
be dodged at every opportunity.

we teachers go on (as we must)
pretending that when we say things like
“an equation” or “the *product* law”
we believe that our auditors
are thinking of things like
“a string of symbols
representing the assertion
that a thing-on-the-left
has the same meaning as
a thing-on-the-right”
or “the rule (in its context)
about *multiplication*”.

but if we ever look at the documents
produced by these auditors in attempting
to carry out the calculations we only
wish we could still believe we have been
*explaining* for all these weeks?
we soon learn that they have been thinking
nothing of the kind.

anyhow. the example at hand.
calculus class is encountered by *most* of its students
as “practicing a bunch of calculating tricks”.

the “big ideas”… algebra-and-geometry, sets,
functions, sequences, limits, and so on…
are imagined as *never to be understood*.
math teachers will perversely insist on
*talking* in this language when demonstrating
the tricks.

well, the “big ideas” that *characterize*
freshman calc are “differentiation” and
“integration”. for example “differentiation”
transforms the expression “x^n” into the
expression “n*x^(n-1)” (we are suppressing
certain details more or less of course).
this “power law” is the one thing you can
count on a former calculus student to have
remembered (they won’t be able to supply
the context, though… those pesky “details”),
in my experience.

anyhow… long story longer… somewhere
along the line, usually pretty early on…
one encounters the weirdly mystifying
*natural logarithm* function. everything
up to this point could be understood as
glorified sets, algebra, and geometry…
and maybe i’ve been able to fake it pretty well…
but *this* thing depends on the “limit” concept
in a crucial way. so to heck with it.

and a *lot* of doing-okay-til-now students
just decide to learn *one thing* about
ell-en-of-ex
(the function [x |—-> ln(x)],
to give it its right name
[anyhow, *one* of its right
“coded” names; “the log”
and “the natural-log function”
serve me best, i suppose,
most of the time]):
“the derivative of ln(x) is 1/x”.

anything else will have to wait.
but here, just as i said i would,
i have given the student too much
credit for careful-use-of-vocabulary:
again and again and again and again,
one will see clear evidence that
whoever filled in some quiz-or-exam
instead “learned” that
“ln(x) is 1/x”.

because, hey look.
how am *i* supposed to know
that “differentiation” means
“take the derivative”?
those words have *no meaning*!
all i know… and all i *want* to know!…
is that somewhere along the line in
every problem, i’ll do one of the tricks
we’ve been practicing. and the only trick
i know…. or ever *want* to know!… about
“ln” is ell-en-is-one-over-ex. so there.

it gets pretty frustrating in calc I
as you can imagine. to see it in analysis II
would drive a less battle-hardened veteran
to despair; in me, to my shame, there’s a
tendency… after the screaming-in-agony
moments… to malicious glee. (o, cursed spite.)

because, hey, look. if we all we *meant* by
“ln(x)” was “1/x” what the devil would we have
made all this other *fuss* about? for, in
your case, grasshopper, several god-damn *years*?
did you think this was never going to *matter*?
in glorified-advanced-*calculus*? flunking fools
like this could get to be a pleasure.

but how would i know. i’m just the grader;
it’s just a two-point homework problem.

and anyway, that’s not really the *exact* thing
i sat down to rant about.

next ish: *more* bad freshman calc from analysis II.

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