Workshop #10
This is likely to be the last workshop writeup assigned this semester. Try to convince us (and, more importantly, yourself!) that you have learned about exposition of technical material. Do a good job.

Please write a solution to the second problem. I was shocked to learn, as several people pointed out, that the series in part a) is related to a homework problem. How is this possible? Part a) can be done using several different techniques. For example, the last part of the Leibniz Test for Alternating Series on page 585 allows an estimation of the error which can be used here. Alternatively, both parts of the problem can be satisfactorily handled using integrals (but if you do this, in part a) you need to address the signs and how that might make things better or worse).

Rick: How can you close me up? On what grounds?
Captain Renault: I'm shocked, shocked to find that gambling is going on in here!
[a croupier hands Renault a pile of money]
Croupier: Your winnings, sir.
Captain Renault: [sotto voce] Oh, thank you very much.
Captain Renault: [aloud] Everybody out at once!

I've received a number of e-mails and had several extended conversations about this workshop problem, especially part a). Here is an e-mail response I sent to one student. I hope this is helpful.

The last line of the Leibniz/Alternating Series Test in the textbook (please look at this, NOW!) exactly states that the sum of a series which satisfies the three criteria of this "test" will always be "bracketed" by odd and even sums. That is, the sum will be between any two successive partial sums. Since these sums are exactly gotten one from another by adding another term, the error involved will always be at most (that is, less than or equal to) the absolute value of the last term.

This makes this sort of series absurdly easy to estimate via partial sums. I didn't discuss this in class because the situation is somewhat artificial, but it does fit the situation of part a) of the workshop problem.

Let me discuss two examples to make clear what I am saying.

For an alternating series example, I could consider 1-1/2+1/3-1/4 etc. (the alternating harmonic series). I know ("theory tells me" -- this will be covered in the next 2 class meetings) that the sum of this series is ln(2), about .69314. However, the last line of the Alternating Series Theorem tells me that S_{100}, the 100^th partial sum, will be within 1/(101) (abs. value of the first omitted term) of the sum. Indeed, in another window Maple informs me that this sum is .68817. And that is CERTAINLY within .005 of the true value, and 5/(1000)=1/(200) is less than 1/(101).

But things are different and much worse for a random convergent series. Let us consider the series whose n^th term is 1/n^(1.5), a p-series with p=1.5>1, so it converges. The sum goes from n=1 to infinity. Now this series seems to have sum 2.61237 (approximately). The sum of the first 100 terms is 2.47113. The 101st term is .000985. So certainly the first omitted term is NOT going to give any sort of error estimate (why should it -- there are a heck of a lot more terms to add up and they DO!). In the case of alternating series, there is steady cancellation from one term to the next, and since the magnitudes are getting smaller, the cancellations all get closer to the value of the sum (the ruler is folding up!). So to get error estimates for "random" series is much more difficult (the two techniques I have mentioned so far are overestimating with geom. series and with integrals).

I hope this explanation helps.

Workshop #8
Please write a solution to the third problem, about Air Resistance. You should work with the letters, please, and not "specialize" with numbers! You may wish to follow the Hint in the textbook.

The assumption in this problem (v´=–g–kv) is actually physically reasonable. Remember that F=ma and in this problem m=1 (just the sort of assumption I like!). The letter a represents acceleration, and is v´. The –g represents gravity pulling down on the projective. The –kv declares that the force of air resistance is opposite to velocity and has magnitude directly proportional to the magnitude of velocity (so if you travel twice as fast in a resistant medium you will have twice as much resistance). I think this is reasonable, and matches experience.

Workshop #7
You should write a solution to the first problem, about a Taylor polynomial and how close it is to the function. You may find the discussion here particularly useful. I wrote the answer to a QotD, and then tried to compare the Taylor polynomial which is the answer to the original function (x^1/3). There are several pictures and a step-by-step error analysis. This is similar to what you should do for this problem so please look at what's written here. In particular, you should be sure to give a reason why inequalities you write are correct. Just writing the inequalities alone is not sufficient.

Workshop #3
I'd like you to write a solution to the third problem from the third set of workshop problems. The computations in this problem turn out to be remarkably physically significant. For example, the limiting value in part c) explains, in a mathematical way, why high-frequencies tend to have less power (vibrations, music, electricity, etc.). You should try to explain this limiting conclusion both algebraically and graphically. This could be done with only a few sentences but try to write effectively.

Further comments (added Tuesday, February 10)
A conversation with Mr. Zaphiros, followed by futher discussion with other students has led to the following suggestions which may be helpful.

• Several graphs are requested in part a). You certainly should use any appropriate technology (such as a calculator or a computer) to help you with this part, and just copy the graphs generated. These pictures are intended to help you with the major parts of the problem.
• If you begin part b) you will notice that the computations are rather repetitive. It may be better (I strongly encourage this!) to do one computation of an indefinite integral with the parameter N, unspecified, first. Then the computations (the specific evaluations) in b) can be done easily (you will report these decimal numbers, please). And you can finish the problem in part c) with the use of the answer with the parameter N. The best explanation will have two parts which reinforce each other. One part will be an algebraic limit computation, and the other part will involve considering what happens to the geometry of the definite integral of the function when N is really large (maybe 10,000?). Hint: the word "cancellation" may be useful.

Workshop #2
I'd like you to write a solution to the fourth problem from the second set of workshop problems. Here are some comments which may be helpful to you.

• Notice that if you are given two consecutive integers, then one of them is even and one is odd. Notice also that in many basic textbooks, most odd-numbered problems have some sort of answer in the back of the …. Ooops.
• In section 6.5, EXAMPLE 3 is entitled Pumping Water out of a Tank and its solution may be related to the solution of this week's workshop problem. Compare the pictures!

Further comments (written Sunday morning, February 1) I've read the work of two of the three sections that I'm looking at this week. First, students who gave no explanations but who gave what can be read as correct mathematical analysis of the problem received 5 points out of 10 (nothing for exposition, and all credit for content). Students who handed in more than one page and who did not attach the pages (for example, by a staple) lost an extra point. Parts a) and b) of the first writeup were totally routine. The situation in c) was a bit novel. Some students handled it correctly and well. Others wrote what seemed to be almost random equations and asserted incorrectly that what was written modeled what part c) described. They were penalized.

Workshop #1
I'd like you to write a solution to the fourth problem from the first set of workshop problems. Here are some comments which may be helpful to you.

• You do not need to "reinvent the wheel" for this workshop. By that I mean you don't need to recite all of the background having to do with the solution of the differential equation dy/dt=Ky etc. This material is discussed in section 5.8 of the textbook. The correct and professional thing to do is to cite specific pages in this section and then assert, with citation of specific formula numbers as needed, that you will use certain equations from this section. Therefore, most remarkably, you should be able to complete this writeup without having to do any calculus yourself!
  But you will need to explain what you are doing with the formulas, and that's a bit intricate.
• Your explanation should be in complete English sentences. It would be nice if there were no (or few) spelling and grammatical errors. In the real world, when a written presentation represents you, this can make a difference, really, really.
• The problem can probably be completely written up in one page. You do not need to show simple and straightforward computations ("2+3=5") but, again, you must explain the ideas. Certainly a solution longer than two pages would be rather exceptional. An answer which is more than one page should have the pages permanently fastened (stapled). Answers should be neat. Again, in the real world, neatness matters.
• It is possible to solve this problem exactly and write a complicated expression with values of logs as the answer. This is acceptable, but, in reality, a useful final answer would usually be accompanied by a phrase like, "and this is approximately [floating point answer, to say, 3 decimal places]". The numerical answer alone is not a satisfactory answer to the problem. You must explain what the answer means in the context of the problem. If your answer is 67.883 (this isn't the correct answer) it should have some contextual information, such as "67.883 micrograms" or "67.883 minutes since ..." etc. This is key to a serious, useful solution of even this artificial problem.
  You do not need to solve this problem exactly if you think you can do sufficiently good numerical approximations at intermediate stages to get an accurate approximation to the final answer. Although this may seem strange, it is always, in real problems, a consideration which can cause anxiety -- I'm sorry.
• You may discuss solution of this problem with other students and with course instructors, but the work you hand in should be your own and anything you copy from a source should have a specific citation.
• You may or may not feel that a graph would be useful in your explanation. If you do include a graph, you should label the graph correctly (including the axes) and refer to it in your explanation.

Maintained by greenfie@math.rutgers.edu and last modified 1/22/2009.