The Calculus Diaries - Jennifer Ouellette [111]
Finally, we can rewrite eC as simply C since both are constants; that lets us combine them into one big constant. After all is said and done, we end up with the function y = Cekx. Whew! This is the equation that encodes the answer to the question we’ve posed.
If you’re anything like me, by now your head hurts, and you just want to be done with it. But remember the zombies! We need to figure out how fast the zombie infection is spreading, because our very lives may depend on it. As we learned in chapter 2, the solutions to differential equations aren’t specific numbers; they are new equations. And this particular equation holds the key to determining the growth rate of the zombie population.
What does our new equation tell us? Well, C stands for the initial population of zombies (a constant number that doesn’t change), while y now stands for the total zombie population after a certain amount of time (denoted by x) has passed. We’ve got Euler’s constant (e) lurking around as well, but we need not worry about it just yet. That leaves k, which will tell us the rate of zombie infection.
We need to figure out the value of k. We start with the initial zombie population: Let’s say on day 1 there were 19 people who ate contaminated hamburger and turned into zombies ravenous for tasty brains. Ten days later, we count again and discover their ranks have swelled to 193 zombies. That’s all the data we need to solve for k, using our handy little formula: y = 193 (new number of zombies), C = 19 (beginning number of zombies), and x = 10 (days that have passed): 193 = 19e10k.
Next we take a series of steps to simplify and solve this equation. Since k is the value we want to find, we must isolate k on the right side of the equation. First we divide by 19 on both sides of the equation: .
Now we need to get rid of that annoying exponential function, which we do by reintroducing the natural logarithm: .
Next we divide both sides of the equation by 10 to isolate .
Finally, we get our answer: k = 0.2318. Aren’t you glad you invested in that calculator? Brace yourself, because we’re not quite done yet.
Now we can plug the values for C and for k into our handy little equation: y = 9e 0.2318x. With this information, we can determine the number of zombies there will be any number of days in the future, just by varying the x factor. Voila! We have a truly predictive model; that is the beauty of the mathematical function.
For instance, how many zombies will there be after thirty days? Make x = 30, and we get 19,914 zombies. Hmmm. That’s some serious exponential growth. I’m sure our merry band of zombie killers would agree: We will be outnumbered very quickly. Evacuation is definitely in order.
BIBLIOGRAPHY
BOOKS AND ARTICLES
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Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time. New York: Thunder’s Mouth Press, 2006.
Bell, E. T. Men of Mathematics. New York: Simon & Schuster, 1937.
Beller, Peter C. “Fill ’Er Up with Human Fat.” Forbes, December 21, 2008.
Berlinski, David. A Tour of the Calculus. New York: Pantheon, 1995.
Berzon, Alexandra. “The Gambler Who Blew $127 Million.” Wall Street Journal, December 5, 2009.
Bland, Eric. “Web-Crawling Program ID’s Disease Outbreaks.” Discovery News, July 18, 2008.
Bohannon, John. “Social Science: Tracking People’s Electronic Footprints.” Science 314, no. 5801 (November 10, 2006): 914.
———.“Friends or Acquaintances? Ask Your Cell Phone.” ScienceNOW, August 17, 2009.
Bressoud, David M. The Queen of the Sciences: A History of Mathematics (DVD). Chantilly, VA: Teaching Company, 2008.
Chang, Kenneth. “Study Suggests Math Teachers Scrap Balls and Slices.” New York Times, April 25, 2008.