When you're forced at gunpoint to write a paper for a non-paper class, no good can come of it...
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", Sanford sneered. 
      "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?" 
      "Calculus Man!" 
      "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 anything?" 
      Sanford took a good look. "Why yes I do. The first one, of the thrown Spam,  is negative. The other is positive." 
      "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. 
 
 

 

The End 
 
 
 
 
 
 
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