Death by Calculus

by Hallan Mirayas



Arrrrrrrgh! I flipped my pencil around and scrubbed out a line of
equations… for the fourth time. 'I hate math!' I growled to myself. That's
when the eraser decided to shear off and roll away under the table while the
rest of the pencil gouged into the page, tearing it. As thoughts of a fiery
doom for my math book filled my head, I smoothed down the rip in my
notebook, trying to salvage a half-hour's worth of limits. Why on earth
Mrs. Mann, calculus demoness from Hades that she was, wouldn't let us use
laptops or even calculators was beyond me, but it translated to a lot of
work on paper.

I ducked under the table to grab my eraser, and wished fervently that
Jubatus were here. I'm sure he'd know how to do this problem.


"Good afternoon--"

I just about jumped out the roof. "Aaaagh!"

It was the plant lady… um, Sue Carter, I think. She arched finely
drawn eyebrows at me and commented dryly, "A bit high-strung today, are we?"

Jubatus had warned me about her, that I shouldn't trust her or talk to
her about him. So I put on the most innocent face I could, and said, "Um, I
haven't seen Jubatus around lately, if that's what you're going to ask."
Then I noticed that I'd dropped the eraser when I'd jumped, and scowled,
looking around for it. Again. "You wouldn't happen to have seen the eraser
I just dropped, would you?" I stooped to look under the table again,
keeping one ear towards her for a reply.

"It went off to your left," she said. "You're Hallan, right?"

I looked left to where she pointed, and found the little bugger next to
the piano. "Yes, ma'am," I replied as I settled back at the booth, careful
to thread my tail through the slot at the back of the seat rather than sit
on it.

"I couldn't help but overhear that you're having some trouble with math."
She gestured toward my much-rumpled notebook. "Would you like some help?"

Setting the eraser down out of the way where it wouldn't get bumped off the
table and lost for a third time, I nodded and answered, "It's introductory
calculus stuff…" I pushed the book over where she could see it. "If that's
not a problem, then no, I wouldn't mind."

She "hmmm"ed for a moment, then slid into the seat across from me and
reached across for my notebook, producing a pen of her own as she flipped to
a clean page. "Differential calculus, at its most basic level, is simply a
mathematical tool to study the ratio of change of events. As an example,"
she quickly drew a square graph and a straight line, "here we have a line
with a constant rate of change, said rate being the value 'b' in y=bx+c.
You've taken that already?"

I nodded.

She drew another graph, this time with a perfect circle as the line. "In
this case the slope is not constant, but the rate of change of the slope is
a constant as this is a circle. In most cases however, the rate of change
of the slope changes, a simple case being the elliptical orbit of Brin

She drew another graph, this time with the line being an ellipse. I was
actually following this!

"Back to Differential Calculus. Right now you seem to be wrestling with the
concept of limits, and probably have no clue as to why you're being stuck
with such a useless task. The key to mathematics is understanding the
reason why things are done. Differential Calculus exists to determine the
slope; and in the case of a straight line the slope is trivially obvious.
With a curved line though the slope varies, and thus you want the slope at a
single point of the line, at a particular set of values, or at a limit of
those values for that line." She drew another graph and this time a curved
line arcing upward.

I half nodded, one ear dropping slowly back. As she went on, she talked
progressively faster…

"The slope of this straight line can be represented by (f(b)-f(a))/(b-a)
where f(b) and f(a) are represented by using the values of b and a
respectively for x."

"Okay..." Where was she going with this?

"Allow the point b to move closer and closer to point a." She drew a number
of lines of increasing slope so that a and b were closer and closer
together. "Eventually, you reach the point where a and b are equal, and the
straight line is the tangent line to the graph of y=x^2 at the value of x=1.
Thus the slope of the line any line at any point can be represented by the
general equation..."

She turned the page and wrote an equation in large print: Lim(x->a)
[(f(x)-f(a))/x-a)] My head started to spin.

"In this case let y=f(x) be a function, and suppose that a is in the domain
of f. In other words, a is a valid value of x within the equation y=f(x).
A line L containing (a, f(a)) is a tangent line to f at x=1 if the slope of
line L is the limit. Thus, your problem is actually determining the slope
of y=x^2 for the value of x=1, what we designated a, or in the way the
problem was given to you, the limit as x approaches 1 for the equation

"Um..." I half reached for my notebook, but she flipped to the next page
and started jotting faster, her eyes locked to what she was doing.

"Now, lets go through a more complex and practical example. Let's say that
Babylon is approaching Brin station along a parabolic curve that can be
represented by..."

"Wait a--"

"Of course Brin is also moving in an elliptical orbit, and it is rotating.
We can ignore quantum variances because the speeds involved are low enough
that Newtonian mechanics still holds. If we want to calculate the distance
between one end of the rotating line that represents Brin, call that point
a, and Babylon, we generate this equation with respect to time, given that
Brin's rotational momentum..."

"Hold on a--"

She turned another page and kept writing. "For large distances, we can
abstract Brin to a point, and represent the distance as a straight line, the
hypotenuse of a right-angled triangle made up of the points a (Babylon), b
(Brin) and c. If we ignore motion, this becomes simple trigonometry.
You've taken Pythagorus?"

I looked on in dismay, giving up on trying to get her attention outside of a
bullhorn to the ear. And I didn't feel like losing my voice roaring at her.
I guess she assumed I nodded because she plowed right on.

"From Pythagorus we get (a-b)^2 = (b-c)^2+(c-a)^2. But, remember that both
Babylon and Brin are moving. If we use a first level mathematical
approximation of their orbits..."

More numbers appeared. And letters. She filled the page and moved on to
the next one.

As she droned on, I looked around in a panic and noticed Jubatus just
coming in. He must have seen the desperation on my face, because he walked
over and carefully pulled the green math demon's hand away from the page. I
heard a rustle of flipped pages and saw that my notebook was back open to
the original problem. "Jubatus, how...?

He ignored me and turned to the plant lady and asked her, "Carter, have
you forgotten what grade the kid's in? This is graduate-level stuff, maybe

I sighed with relief as she turned away from my book towards Jubatus.

"This would have been no problem for me at his age," she answered him.

"Excuse me..." I tried to interject.

Jubatus answered her, "You were in college at age 15?"

Sue glared at Jubatus. "I will admit that I was a little advanced for
my age, but I had thought that the North American educational system still
had some quality left within it."


He shook his head in apparent disbelief and then turned back to me. At
least he had heard me. "Okay... what you want is a series of successive
approximations. Start out with what looks like a decent guess, run that
thru the equation, and if that doesn't get the exact answer, try some other
guesses. Say you start out with 1.9, and it turns out to be a little low;
the next guess might be 1.95. And so on. You want to look for what the
values are trending --"

I started scribbling notes but she didn't let him finish, "Here we have
the modern American way. Keep them in the dark and make math a simple
mechanical tool. So tell me Mr. Jubatus, WHY'D YOU KEEP YOUR INJURY
SECRET?! I could have helped you."

I flinched away from her shout, which drew looks from every other
patron in the bar. By this time, I'd given up on Carter entirely and
focused all my attention on getting Jubatus to finish his explanation.
Meanwhile, they continued their rapid back-and-forth sparring.

He turned away from me and answered, "Carter, if someone asks you what
time it is, you don't give them the blueprints for a bloody atomic clock!"

The plant lady turned her level of sarcasm up, "And if they don't know
WHY an atomic clock works, why not just tell them what time it is?"

"Exactly," he retorted, the fur along the back of his neck rising.


She turned to me and sharply stated, "The answer is 2," before turning
back to Jubatus. "I was concerned about you! If you'd have died--"

Jubatus cut her off. "-- it would be the same thing that happens to
everyone, sooner or later."

"Jubatus, could you repeat what you said...?"

She plowed into him with barely a pause for breath, "Do you have ANY
idea what the cost to hum--"

Apparently Jubatus had heard me, "So. Successive approximations. Does
that make sense to you?"

"I think so, but --"

Sue bulled her way back into the conversation, and Jubatus' twitching
tailtip suggested he wasn't going to stand that for much longer. "Mr.
Jubatus, you will listen to me now. If I was present you would have
recovered your mind faster. You know it; I know it. Since you want to keep
the subject sec--"

"Jubatus..." I had a sinking feeling as to whether they were ever
going to stop arguing, but I held onto one slim hope: If I could just
squeeze a few more clarifying sentences out of --

Jubatus turned back to her and snapped, "Enough. You want to discuss
this, we do it in private. Back room. NOW." Then he turned and stalked off,
Sue hot on his tail.

"Jubatus!" I called, too late to do more than have it bounce off the
door that slammed shut behind the pair.

"Well, that was helpful," I grumbled as I picked up my pencil. I
thought she'd said the answer was 2, and Jubatus had said something about
successive approximation... Whatever 'successive approximation' was. Sigh…
Back to figuring this out the hard way… Open book to page 12, and let's go
over this again...





Copyright 2002 by Hallan Mirayas. If you want to post this anywhere else, please ask for permission first. Thank you.

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