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;
instead it provides links to w’edia pages on each).

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
already covered in the prerequisites-so-far).

if i recall correctly, my own experience of
learning-about-e was rather a horrible mishmosh
of formal-correctness and we’ll-learn-about-this-later.
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.


  1. When I teach calc I (it’s been a while!), I work through the derivative of 2^x with them. We have a mystery number as part of the derivative (this number is the slope at x=0). Once in a while, someone recognizes .69, but mostly we just keep it around for a bit. I have them do 3^x, and we get a new mystery number, 1.1. We get if y = a^x, then y’ = k*a^x, and k increases as a gets bigger. So we invent e, and then we can find ln2 and ln3 from that, and clean up the others.

  2. blag

    somebody *recognizes* .69
    (as an approximation to ln(2))?
    wow. *i* didn’t recognize it!

    anyhow, this certainly looks like
    a reasonable way to motivate the
    “exponential whose slope at (0,1)
    is 1” approach to defining the
    natural-exponential function.

    *if* we had world enough and time.
    at least an hour lecture on this one topic.

    i’ve seen this approach in texts,
    of course. never had a chance
    to do the lecture this way though.

    if there’s an intro-calc course
    with this kind of leisurely pace,
    *i’ve* sure never seen it.

    many of the students are going to believe
    that the whole “derivatives are limits
    of difference quotients” thing is just
    a bunch of mumbo-jumbo anyway.
    “just show me the formulas;
    never mind why”.

    if i were to design the course, everybody
    would darn well have to find derivatives
    by taking limits-of-DQs even if that meant
    taking two weeks to do linear functions.

    of *course* students who don’t know
    how and why to “cancel” the delta-x’s
    and when to quit writing “lim”
    for *easy* functions aren’t somehow
    suddenly going to catch on by watching
    it done with *tricky* functions…

    but, hey. look here. that section was
    “covered” last week. too late! too bad!
    better just “memorize” something and pray.

  3. Maybe my memory’s bad – maybe no one does ever recognize it, I’m not sure…

    Our course is definitely not leisurely, but this seems an important point, and I take the time. I go too fast over lots of other stuff. (A large component of the art of teaching well within a school situation may be knowing when to slow down and when to speed up.)

    In my intermediate algebra class, I tried to go too fast through the rational expressions and roots, so I’d have more time for my beloved exponential functions. But the students asked lots of good questions and slowed me down. We’ll still do the murder mystery in the exponents unit, but something will have to give at the end. I may skip inequalities entirely (along with conics and series).

  4. i quite agree about the “large component”.

    the selection of topics for our courses,
    and their textbook presentations,
    are determined by committees
    (and corporate entities even less
    interested in human needs).
    “historical forces”… inertia mostly…
    play a large part too.

    evolution… *not* intelligent design.

    anyhow, after one has presented
    a course by following the stated rules,
    one will have learned quite a bit
    about where the most useless timesinks
    are and what essential material is
    being mentioned and passed over
    much too quickly (or ignored altogether).

    if i’m right in thinking that my students
    are generally lucky to have me as a teacher,
    my ability… and willingness… to use my
    judgement on these matters might very well
    be the best single reason.

    some of the other teachers aren’t able;
    i hate dissing math teachers, so enough
    about that. many, maybe most, of the rest
    aren’t *willing*… and no damn wonder.
    you’ll be punished if you try.

    much safer to just say “hey, i’m doing
    my job” like any other government drone
    refusing to help the people they’re
    supposedly serving because the important
    thing is to appear to be following rules.

    enough. thinking about working conditions
    instead of actually working has made me
    a nervous wreck. (that and things like
    “no paychecks this quarter”.)

    i’m hoping it’s the *graphical* component
    of “inequalities” you skip. everybody needs
    to know *something* about ’em…
    namely “change signs and directions
    together or not at all”.

    then they throw in absolute values
    and botch the job badly; along comes
    *graphing* inequalities and they botch
    the job even worse.

    the final exam at my ex-community-college
    gigs had a “graph the inequalities” problem
    wrongly solved on the answer key
    for years. probably still does.
    and no being with the authority to fix it
    has any interest in the truth.

    conics for precalc is also well worth skipping
    (if only one could). i first saw ’em in calc ii
    and got nothing out of it but confusion.

    too bad, too. if geometry hadn’t been
    sidelined somehow this could be great stuff.
    conics are closely related to all this drawing
    i’ve been doing
    . but the very math *majors*
    typically learn nothing or almost nothing
    about projective spaces alas. and waving one’s
    hands at the topic in the form of “memorize
    a whole bunch of formulas you’ll never need
    again” would obviously be a bad idea if one
    were actually to work with a few classes
    that’ve been made to try it this way.

    series? not sure what you’re getting at.
    sometimes… business calc, e.g. …
    one sees “sigma” notation introduced
    in contexts where about three weeks
    worth of material is to be presented
    in a day. my recent “discrete” class
    did this too. one can do the lecture
    of course but *only* students having
    considerable prior exposure are likely
    to get anything *out* of ’em.

    somewhere along the line, though…
    calc iii or so typically… they’ll become

  5. What I’d have done with series in intermediate algebra is show them how mortgage payments are calculated, and how interest accumulates. I convinced students back in Michigan to never take out 30 year mortgages.

    We hit very basic inequalities as we talk about domain, and I’ll be using that as my context if we have time for our inequalities unit.

  6. good discussion for me to read! Thanks and thanks.


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