### bad freshman calculus

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

.

(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.

## Leave a Comment