Calculus Man Saves The City
Crash! Another can of Hormel Spam had been lobbed through another
window. This was the fifth hostile Spamming in as many days. Who was throwing
these cans of Spam? Where were they coming from? When would the fine citizens
of Math City be once again safe from these unknown Spam-throwing terrorists?
Who will save them?
Look, up in the sky, itís wearing Spandex,
itís a bird, itís a plane, no, itísÖCalculus Man! "Never fear, fine
citizens of Math City. I will use the mystical powers of Calculus to uncover
the perpetrator of all this flying Spam. Your city will soon be free of
this scourge. To bring this Spam-throwing scumbag down, I will need to
prove his guilt. Calculus will help us do this. Calculus will triumph over
evil!" With that, Calculus Man reached into his pocket and whipped out
Challenger, his all-knowing TI-85.
With a loud Boom!, a tin of Spam had
been launched from a Spam Cannon at an unknown location. "Sailing Spam
at four oí clock!", cried one of the bystanders. Sure enough, yet another
can of the spiced-ham product was headed straight for a large plate-glass
window. Using his exceptional powers of observation, Calculus Man determined
the exact position of the can at two very close time intervals, and the
time between launch and impact with the window. "Quickly, Challenger, get
me a derivative!" Calculus Man knew that the derivative would tell him
the velocity at which the Spam was moving. This, combined with the time
it took to crash through the window, would be enough to determine the exact
distance that the Spam had traveled, 2.5 miles. He had also determined
the Spamís height at various points during its travel, and calculated a
derivative for this height as well. Drawing a graph of the derivative of
the Spamís height, Calculus Man called it to the attention of the bystanders,
who were utterly awed by his knowledge. "Look here, fine citizens. Notice
how this derivative graph is positive, then becomes negative. Where this
sign changes there is a local maximum. That means the Spam reached its
maximum heightÖright over city hall, which is due east. The Spam launcher
is 2.5 miles east of here.
Calculus Man leaped over several tall
buildings in a single bound to land at the location of the launch. He then
approached the launcher, a fat, ugly villain named Sanford. "You think
I threw those Spams, donít you, math-boy?!? Iíd like to see you prove it",
"Gladly", he said. "And thatís Calculus
Man to you." He then went on to explain how he found Sanford with the two
derivatives, just as he had explained it to the citizens.
"Yeah, well, that doesnít prove that
I did it, now, does it, mathelete?"
"Right, whatever. Now suppose that instead
of being launched from a Spam Cannon, these cans were instead swung
through those windows?" Sanford, twisting his mustache in his fingers,
drew a diagram showing a man in a hot-air balloon, dangling a can of the
Hormel food product on a rope. "Couldnít an airborne miscreant just as
easily have gotten this rope swinging, then just let the Spam crash through
the window that way? Iím sure thatís what really happened."
"Interesting hypothesis, Sanford, but
unfortunately it does not hold water." Calculus Man sketched a graph of
a can of Spam flying through the air as if thrown, and another of a can
flying as if swung on a rope. He then sketched the derivative of each.
Finally, he sketched the derivatives of the derivatives he had just drawnó2nd
derivatives. "Look carefully at these 2nd derivatives. Notice
Sanford took a good look. "Why yes I
do. The first one, of the thrown Spam, is negative. The other is
"Very good. Do you know what that means?
When the second derivative is negative, the first derivative, the rate
of change, is decreasing, This means that the function is concave-down.
If you poured water on this function it would run off. But when the 2nd
derivative is positive, the first derivative is increasing. The function
is concave-up, and if you poured something on this function it would hold
it. If the Spams were swung through those windows, (points again
to the swung-spam graphs) a graph of their height would be concave-up,
right? But look. What we encountered today was definitely a thrown
Spam." Calculus Man turns again to his initial graph of the Spam that led
him to Sanford and its derivative, and sketches out the 2nd derivative
here also. "That 2nd derivative is negative. This function is concave-down.
This Spam was thrown and not swung. Your story, much like the function
of your Spamís height, literally does not hold water."
Lights. Sirens. A squad car pulls up
and two heavily-armed officers step out. "Calculus Man, is this the guy?"
"All right, Officers, I confessóI launched
those cans of Hormel Spiced Ham through your windows. I terrorized Math
City with those Spams. It was I! And I could have gotten away with it too,
if it wasnít for Calculus Man. And that stupid calculator!"
Sanford was cuffed and taken to
court, where his punishment was decided: He was condemned to replace the
windows that had been broken and eat each and every Spam he had thrown.
The fine citizens of Math City were safe from evil once again.