Archive for the ‘Math 151’ Category

$log_a(x) = ({{ln x}\over{ln a}}) = {1\over{x ln a}}$ 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)].)

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*
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)

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
“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
your case, grasshopper, several god-damn *years*?
did you think this was never going to *matter*?
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.

blag.

two rungs make a rite

logarithmic differentiation at $GL(s, \Bbb R)$.

MEdZ 0.4 debut

at long last. this has been sitting
on the paperpile very-nearly-finished
for quite a while.

i started the “lectures without words”
series early on with 0.1: $\Bbb N$.
whose cover more or less announced
implicitly that it was one of a series
called $\Bbb N \Bbb Z \Bbb Q \Bbb R \Bbb C$. and that
was, like, five quarters ago.
and they’re only 8 micro-size pages.

a couple days ago i inked the graphs
and the corresponding code (the stuff
under the dotted line had *been* inked
and the whole rest of the issue was
entirely assembled). and zapped it off.

and yesterday i passed ’em around
at the end of class (to surprisingly few
students given that i’ve got freshly-

$\Bbb Z$ i did right away
(if i recall correctly), and in
high-art style, too (i used a
$\Bbb Q$ wasn’t much later.
i have plenty of notes for $\Bbb C$,
too, and could knock out a version
on any day here at the studio
(given a couple hours and some
peace of mind) that’d fit right in.

anyhow, what we have here are,
first of all, obviously, a couple graphs
and a bunch of code. here, at risk
of verbosity, is some line-by-line
commentary.

in the upper left is
part of the graph of
the linear equation
y=(x+1)/2…
namely the part whose x’s
(x co-ordinates) are between
-1 and 1.

and my students (like all
are familiar with *most* of the
notations… and *all* the ideas…
in this first line.

that funky *arrow*, though.
well, i can’t easily put it in here
(my wordpress skills are but weak)
but i’m talking about the one
looking otherwise like
$(-1, 1) \Rightarrow (0,1)$.
and in the actual *zine*, it’s
a Bijection Arrow.
something like ” >—->>”.

as explained (or, OK, “explained”)
*below* the dotted line.
where *three* set-mapping “arrows”
are defined (one in each line;
the $:\Leftrightarrow$ in each line
denotes “is equivalent-by-definition to”).

the Injective Arrow >—> denotes that
f:D—->R is “one-to-one” (as such
functions are generally known
in college maths [and also in
the pros for that matter; “injective”
and its relatives aren’t *rare*,
but their plain-language versions
still get used oftener]).

“one-to-one”, defined informally,
means “different x’s always get
different y’s”. coding this up
(“formally”), with D for the “domain”
and R for the “range” (though i
prefer “target” in this context
when i’m actually present to
*explain* myself) means that
when d_1 and d_2 are in D,
and d_1 \not= d_2
(“different x’s”), one has
f(d_1) \not= f(d_2)
(“different y’s”).

likewise the Surjective Arrow —>>
denotes what is ordinarily called
an “onto” function:
every range element
(object in R)
“gets hit by” some domain element.

and of course the Bijective Arrow >—>>
denotes “one-to-one *and* onto”.

there’s quite a bit of symbolism
here that’s *not* familiar to
typical college freshmen.
the Arrows themselves.
$\forall$ “for all”
$\exists$ “there exists”
$\wedge$ logical “and”
and the seldom-seen-even-by-me
“such that” symbol that, again,
i’m unable to reproduce here
in type.

still, i hope i’m making a point
worth making by writing out
these “definitions without words”.

anyhow… worth doing or not…
it’s out of the way and i can return
to the main line of exposition:
the mapping from (-1, 1) to (0,1)
in the top line of my photo here
is a “bijection”, meaning that it’s
a “one-to-one and onto” function.

for *finite* sets A and B,
a bijection f: A —-> B
exists if and only if
# A = # B…
another unfamiliar notation
A and B have
*the same number of elements*.
we extend this concept to *infinite*
sets… when A and B are *any*
f:A—>B, we again write
#A = #B
but now say that
A and B have the same cardinality
(rather than “number”;
in the general case, careful
users will pronounce #A
as “the cardinality of A”).

i’m *almost* done with the top line.
i think. but there’s one more notation
left to explain (or “explain”).

the “f:D—>R” convention i’ve been
using throughout this discussion
is in woefully scant use in textbooks.
but it *is* standard and (as i guess)
often pretty easily made out even
by beginners when introduced;
one has been *working* with
“functions” having “domains”
and “ranges”, so fixing the notation
in this way should seem pretty natural.

but replacing the “variable function”
symbol “f” by *the actual name
of the function*
being defined?
this is *very rare* even in the pros.
alas.

the rest is left as an exercise.

tutor room first day

two hours fly by like nothing… working exercises
with one student at a time is generally the best part

there’s quite a bit of homework for 151 students
to work out this week… and then to digitize.
putting the answers into the computer interface
can be expected to present *lots* of problems.

but this level of coding is easy enough for *me*,
so part of how i was able to help with
in a couple of instances was translating.
handwritten work leading to $x \not= 0,\pm { {3\sqrt{2}}\over{2} }$
might be entered as
(-I, -3*2^.5/2)U(-3*2^.5/2,0)U(0,3*2^.5/2)U(3*2^.5,I)
—which is anyway much more trouble to type
than the handwritten version is to write out.
correct code; maybe this is easier than i think.
we’ll see how it goes i guess.

PS
first quiz.
my latest rant mentioned the reason
this was postponed from tuesday to thursday
(or from T to R as i’m likely to use in handwriting):
computer security snafu. so i got a hardcopy on W
and created two slight variants in $\TeX$.
got those run off in the printing office.

and i spread ’em around around the room
with the allotted 15 minutes left to go
(versions alpha and beta to alternate columns).
took the roll for the first time ever and copied
names into “seating chart” order in my notes;
an old habit. everybody handed in with time
to spare; good.

i haven’t graded ’em yet (more or less of course).
but i glanced at a few of the papers and saw
*lots* of right answers including those on
at least one “perfect” paper. looks like it’ll be
“so far, so good” when i’m passing ’em around.
here’s hoping.

blogging calc i

without wanting to commit myself… here goes.
i’m *undercommitted* this quarter goodness knows.

my calculus blogging from spring ’09
might come in handy (but the Calc III stuff
is mixed in with the Calc I).
for that matter, the common errors page (not by me)
that i cited yesterday is *bound* to come in handy.

heck. learning math on the web?
just like learning anything else,
there’ll be plenty of good info… more than enough
to build a course around…
in the ever-amazing wikipedia. let’s see.

consider this list of calculus topics.
hmmm. it doesn’t refer, specifically and directly,
to the topics we looked at on tuesday…
exponential functions, inverse functions, log functions…
but *does* link to the precalculus page that
*does* treat of these topics specifically (though not directly;

continuing in this vein, i’ve just looked (for the first time)
at the exponential function page. hmm.
i imagine myself a beginning student. what do i see?
forest-and-trees issues abounding… there’s an *awful lot*
of material here! but maybe it’s clearer than the textbook
even so. in particular, the article is (very rightly) about
*the* exponential; of course i mean the one with base “e”.
and, right out front, in the first sentence in fact, they’ve got
“the function e^x is its own derivative”.

every calculus *teacher* understands, at least to some extent,
that the importance of the number “e” is very closely tied to
this property… it’s the base that makes exponentiation
“work out conveniently” in “doing calculus”.
most of ’em, if pressed, would probably be able to tell you
that “y = e^x” is the only non-trivial solution
to… the world’s simplest interesting differential equation…
${{dy}\over{dx}} = y$. (the trivial one is y’ = 0).

every calculus *book* obscures this point to some extent.
and there are reasons. one has not *defined* ${dy}\over{dx}$, after all.
*our* text—”stewart”—follows the usual pattern
of “discuss exponentials generally first”
(y = A^x for A a positive
number different from 1) and then singles out the case A=e
as the one having a tangent of slope 1 at its y-intercept.
this can hardly be very motivating for a beginner.
and, anyhow, neither has “tangent” (to a curve at a point)
been defined… so (as far as i can see) *nothing* is gained
in terms of “formal correctness” by focusing on this
particular *detail* of the fact that the exponential
is its own derivative.

okay. there are *better* reasons. textbooks *should*
review exponential-functions-generally (and provide
lots of practice problems). nobody’s going to
understand very much about y=e^x that doesn’t
know anything about its first cousin y=3^x.

still it seems to me that
somewhere pretty close to the moment that
the hugely-important constant “e” is introduced,
it would be helpful to at least some students…
students like i imagine myself to have been,
for example… to have some *succinct*
and *correct* justification (even if its details
can’t be spelled out fully with the concepts

if i recall correctly, my own experience of
learning-about-e was rather a horrible mishmosh
i *did* learn about it later but it was an accident of history;
if i’d merely been a math *major*, it’d’ve been obscure
to me all my life, but since i went on to be a
*graduate student* in maths, i eventually
considered myself duty-bound to make sense
of it all (and had the “mathematical maturity”
to do ahead and do it).

it makes good sense *formally* to consider
the natural-logarithm function (“ln”)… defined
(of course!) as the integral-of-the-reciprocal…
*first*, and define “e” as the solution to
ln(x) =1.

hey, madeline just woke up. more later.

the natural log is *not* the reciprocal

i’ve got a post for spring quarter. yay!
less money again and i suppose less
prestige… i’m running two sections
of a calc class. the students see lectures
(not by me) on mondays and wednesdays
and meet me for guided-problem-solving
(and to hand in homeworks and take quizzes;
exams have yet a different schedule;
i grade everything of course) tuesdays
and thursdays. also there are tutor-room duties.

but one-on-one tutoring is in many ways
the *best part* of the whole “math teacher”
thing. not by coincidence, the furthest away
from the show-me-the-money side:
grant applications and suchlike no-math
in curriculum design and suchlike policies:
worse-than-meaningless committee work.

there’s a great deal of online activity
associated with teaching at Big State U.
but almost everything has gone pretty well
in my two quarters here so far (this century).
the payroll-related interface was frustrating
to be sure… but there was an old-fashioned
paperwork over-ride that i eventually found
to beg for help.

otherwise… filing grades and scheduling tutoring
duties for example… stuff works the first time
and the right buttons aren’t hidden among
dozens of useless ones with similar names.

• (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)