DIFFERENTIATION THEN INTEGRATION
 
 
     C T was walking past one of those massively covered
bulletin boards which were, after all, put in the hallways
for the purpose of being massively covered, when he saw a
photocopied advertisement.  It read:
 
     TUTOR NEEDED:  CALCULUS (Math 119)
     4 Hrs. Per Week
     Good Pay
     4087-239 Ask for Chuck
 
C T felt a need to validate himself for social and academic
purposes on campus in ways other than merely receiving good
grades and building towards that ubiquitous BS in Physics
that was still several semesters away.  Besides, he had no
idea how good "good" pay might be.  He certainly knew the
material.  College placement exams after high school calculus
prevented him from ever knowing the extent of Chuck's
difficulty.  Later in the evening, he called the number.
 
     "Could I speak to Chuck," C T asked in the businesslike
way he was surprisingly able to bring about in his voice.  It
could be Chuck and the negotiations might be underway.
 
     "Just a minute."  There was a bit of faint rattling.  It
wasn't a small room in a dorm from the sound level.  "He'll
be here in a minute."  C T prepared himself for his
"interview."
 
     "Chuck Harrison."  Chuck finally stated.
 
     "Yes, Mr. Harrison, I saw your sign for tutor in Leibniz
Hall today..."  He certainly had, after all.
 
     "Yes, I'm having a really hard time in Math 119.  What
level are you?"
 
     "I'm a sophomore.  I had calc in my high school college
prep.  I got an 'A' without a problem."
 
     "Well I could certainly use someone like you.  I need
about four hours help, like the sign said, and I'll pay you
$5.00 an hour and give you a $50.00 bonus if I get a 'B' or
better in the class.  Does that sound good?"
 
     C T had never known what had sounded "good" until then--
this was a chance to be earning during the year, too!  He
still kept himself academically restrained, however.  "That's
fine.  Where will we meet?  I live in Lodge Hall in the
Valley Area."
 
     "Well my apartment is way over on the other side, near
the Stadium.  There are plenty of empty rooms near the Diag
open evenings.  Can you make it at 19:00 - 21:00 on Tuesdays
and Thursdays?"
 
     C T could make it any day of the week that time.  He
hadn't developed extracurricular activities to the point of
some people he knew.  "I can do it.  Let's find each other in
front of Hemingway Hall tomorrow at 19:00.  That's when we'll
start.  Believe me, we have to start soon.  The midterm will
be a killer!"
 
     Darkness was descending on the October campus when C T
saw a lone figure next to a Macedonian column of Hemingway
Hall.  As he approached, Chuck saw his tutor had arrived.
They shook hands, two somewhat dissimilar hands belonging to
dissimilar people.  C T stood 100 millimeters shorter than
Chuck and carried a similarly reduced bulk.  Chuck was
probably involved in athletics, but then he could simply be
what he is.  C T had learned the shunning of prejudice long
ago.  It yields biased views that have no place in science.
 
     "Hemingway might have something for us," C T proposed.
"I think the door is open, isn't it?" In his lesser days, he
might have imagined a man of Chuck's stature being able to
open the door regardless of the state of the lock.  They
found a multipurpose academic cubbyhole of the kind the
University found useful.  Inside was a table and an array of
desk-chairs.  Hung from the ceiling was a television monitor
used for remote video-classes.  This was a classroom only in
that something esoterically defined by computer registration
as a section of "lecture" was supposed to "meet" here.
 
     "What topics are you covering, Chuck," C T asked as they
sat down and Chuck unpacked his leather briefcase.
 
     "The antiderivative and the fundamental theorem of
integral calculus."  His face worked hard to utter that
string of words.  C T tried his best to start earning the
night's ten dollars and his potential share of the fifty.
 
     "You know what the derivative is pretty well, don't
you?"
 
     "It's the slope of the line."  That was the best he
could do.
 
     "Exactly, and when it's a curve, it's the tangent at
that point.  Now what they're saying in antidifferentiation
is simply, 'if you give me the derivative, I'll give you back
what function it is be the derivative of.'"
 
     "Well that's what our teacher said, but I need some
examples.  I have a big "suggested" list he gave me after
class.  We can do a few now."
 
     They did a total of five problems, to be exact, and C T
learned a lot of calculus.  It atrophies, at its root, when
it is put merely to applications.  Chuck needed constant
convincing and imaging, imaging C T was a natural at.  "My
favorite illustration of derivatives and integrals is this.
Now the speedometer on a car tells you how fast the car is
moving.  That's how fast the odometer, which tells you how
far you've gone, is turning.  It's the derivative with
respect to time.  Now imagine you give me the speedometer
reading.  That is, you give me the derivative.  Now I ask
you, if the speedometer is the derivative of the odometer,
what is the function that the speedometer is the derivative
of--what is the antiderivative of the speedometer, in other
words?" C T was talking with his hands.
 
     "Well if the speedometer is the derivative of the
odometer and you ask what it is the dervative of, then I'd
have to say the odometer."  Chuck's face achieved the first
true glow of accomplishment of the night.
 
     "Congratulations, Chuck, you're right!  Keep that
thought in your mind and every time you drive a car, you'll
see an example of differentiation and integration."
 
     "That's a pretty heavy idea, but I know you know your
stuff."  He pulled out his wallet and drew two fives.  "See
you on Thursday night for sure!"