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).

Spook Speaks

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

yn = yn+1 - yn.

Suppose yn is proportional to both yn and t:

y0 = kt y n,

where k is the growth rate. (You may neglect the death rate in your calculations).

Give an expression for y 1, the total number of tripples after one interval t, in terms of y0, the initial population at t = 0.

Give expressions for y2, y3, and yn, the total number after 2, 3, or n intervals respectively, in terms of y0. Make your expressions as compact as possible and factor out y0. Rewrite your expression for yn in terms of the total elapsed time t by setting t=( t/n).

Now expand the expression for yn you found in part (b) using the binomial theorem. In order to get better resolution and accuracy, let the time scale of your analysis become finer and finer; that is, let n--> as t --> 0. Assume that you can take this limit term by term; your result is an infinite series.. Please write this series.

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

How do you know you can calculate the answer to arbitrary accuracy? For example, how far off would I be if I just added up the first hundred terms? The first N terms? Give me an upper bound for the error after 100 terms, and the error after N terms, Spook, and then prove that the error goes to 0 at N goes to

Now wipe that superior smile off Spook's face by showing him what can be done with Earthling calculus.

Start with the same expression for the increment, y= (kt)y, as Spook used in part (a), and convert it into a differential equation by taking the limit as t --> 0 and using the definition of the derivative. Solve this equation by separation of variables to find the function y(t), which gives the population at time t in terms of the initial population y0. It is your turn to smile. (Spook had to run to the ship's library to look up separation of variables).

Spook's Comeback

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.'

Differentiate my series term by term with respect to time and see if the answer isn't k times my series. Therefore, my series does satisfy your differential equation, since its derivative is k times itself. Now compare your function from part (e) with your answer in part (c). Your function and my series must be one and the same!

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."

Decide between Scooter's and McCool's strategies. Which is more effective in the short run? In the long run? Convince them of your answers.

In the future, will these two treatments ever result in the same number of tripples again? When?

Spook Speaks

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

yn = yn+1 - yn.

Suppose yn is proportional to both yn and t:

y0 = kt y n,

where k is the growth rate. (You may neglect the death rate in your calculations).

Give an expression for y 1, the total number of tripples after one interval t, in terms of y0, the initial population at t = 0.

Give expressions for y2, y3, and yn, the total number after 2, 3, or n intervals respectively, in terms of y0. Make your expressions as compact as possible and factor out y0. Rewrite your expression for yn in terms of the total elapsed time t by setting t=( t/n).

Now expand the expression for yn you found in part (b) using the binomial theorem. In order to get better resolution and accuracy, let the time scale of your analysis become finer and finer; that is, let n--> as t --> 0. Assume that you can take this limit term by term; your result is an infinite series.. Please write this series.

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

How do you know you can calculate the answer to arbitrary accuracy? For example, how far off would I be if I just added up the first hundred terms? The first N terms? Give me an upper bound for the error after 100 terms, and the error after N terms, Spook, and then prove that the error goes to 0 at N goes to

Now wipe that superior smile off Spook's face by showing him what can be done with Earthling calculus.

Start with the same expression for the increment, y= (kt)y, as Spook used in part (a), and convert it into a differential equation by taking the limit as t --> 0 and using the definition of the derivative. Solve this equation by separation of variables to find the function y(t), which gives the population at time t in terms of the initial population y0. It is your turn to smile. (Spook had to run to the ship's library to look up separation of variables).

Spook's Comeback

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.'

Differentiate my series term by term with respect to time and see if the answer isn't k times my series. Therefore, my series does satisfy your differential equation, since its derivative is k times itself. Now compare your function from part (e) with your answer in part (c). Your function and my series must be one and the same!

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."

Decide between Scooter's and McCool's strategies. Which is more effective in the short run? In the long run? Convince them of your answers.

In the future, will these two treatments ever result in the same number of tripples again? When?

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