You are the captain of the Starship Exitprize and tripples have invaded the hold of your ship. You are concerned enough to ask Spook, the Vulgar, for an analysis As usual, Spook takes a superior attitude. Vulgars are experts with infinite series, although they never developed calculus. Hoping to teach you something, he gives his report in a series of riddles, to which you must supply the answers. Supply answers to questions (a) through (d) below, and then go Spook one better by showing him how Earthlings would use calculus to solve the problem in parts (e) through (h).
Let y0 be the population of tripples at some initial time t=0, Captain. The let yn be the population after n intervals of length t have elapsed, i.e., at t=n t. If you allow one more interval to elapse, the population will have increased during the next interval by
Suppose yn is proportional to both yn and t:
where k is the growth rate. (You may neglect the death rate in your calculations).
This series is your answer for the population at time t, Captain. you may calculate the answer to any desired degree of accuracy by including enough terms from the series.
Captain Kork's Comeback
Now wipe that superior smile off Spook's face by showing him what can be done with Earthling calculus.
Captain, I claim that your solution is no different from mine. You have simply renamed my infinite series as your function because Earthlings cannot handle the concept of infinity. Allow me to demonstrate, Captain. In a book in the ship's library, I came across something you Earthlings call the 'derivative.'
Kork's Second Comeback
Now necessarily, Spook. the same differential equation may have many different solutions. Allow me to demonstrate...
About this time, McCool and Scooter burst into your cabin. Scooter says, "You gentlemen can debate theory 'til you're blue in the face, but that does'na help us with our problem. I'm prepared to use radiation from the ship's engine just one time to exterminate 99% of the little buggers right now. Just give me the word, Captain!"
McCool replies, "I cannot condone such a waste of life, Captain. I've got a drug that will cut their growth rate k in half. I think you'll find this a more effective and less brutal method than the one suggested by our engineer."