Reflection on The Differences and Similarities between Calculus According to Newton and Calculus According to Leibniz
Newton and Leibniz are both generally credited with having invented calculus - except by their strongest supporters, who argued that each invented the subject and the other did hardly anything new at all. But to what extent did they invent the same subject? One might take the view that they each invented a different subject, and their separate inventions were merged into a single subject by Euler, who drew different ideas from Newton and Leibniz. What are the differences and similarities between the thinking of these two great mathematicians?
By now, you have read enough about the contributions of Newton and Leibniz to the development of calculus to consider the questions of the differences and similarities between their ideas in very specific terms. For instance, when you consider their thinking about the concept of a tangent line: are their approaches that different; or are they saying more or less the same thing? Or, what about their views of the integral, or of the notion of a curve?
Choose one particular topic, like tangent line, integral, or curve, and specifically describe what the differences are between their approaches and what the similarities are. Would you say that, in some sense or other, that the differences are significant? Or were they more or less thinking about the same thing?
This writing need not be formal, and you don't need to give complete references. (However, if you directly quote from our text or another source, do use quotation marks and mention the source.) Just try to get your ideas across in any way you can. If it makes things any easier for you, invent someone you know as an audience for your thinking. For instance, what about someone not in this class who wonders about the invention of calculus, or even a high school math teacher who wonders what you are learning in your math classes at UW.
You should probably write at least a page, but no more than about two pages (single-spaced, with an extra line between paragraphs) on this topic. But for this essay and all writing for this course, it's the weight that counts, not the length.
Reflection on Euler's Contribution to Calculus
The usual argument in connection with the invention of calculus is about whether the subject was invented by Newton or Leibniz. But one might claim that what these two giants gave us is two separate sets of insights into the solution of certain outstanding problems, and that calculus as we know it today did not really appear on the scene until sometime in the middle of the 18th century as the result of the work of Leonhard Euler and his contemporaries.
Your task in this assignment is to argue for this claim - whether you believe it or not. What were Euler's major contributions to calculus as we know it? What difference did these make? The first calculus textbook was written by the Marquis de l'Hospital in 1696. (He's the one the Rule about the limit of a quotient is named after.) It was based primarily on the work of Leibniz. Euler wrote calculus books on calculus 50 or so years later. What would we find in Euler's books that we would not find in l'Hospital's? Does this make a big difference? What if a textbook had been written in 1720 on the calculus of Newton - how would Euler's have differed from it?
In order to write about the importance of the work of Euler, you don't have to diminish the importance of Newton's or Leibniz's work. You just have to tell how the sets of ideas they gave us would not have, in themselves, served as complete subjects for all the uses we want to put calculus to.
You should write about 3/4 to 1.5 pages (single-spaced, with an extra line between paragraphs) on this topic (with the same comment about length versus weight as above).
Reflection on the Importance of the Rigorous Reform of Calculus from Cauchy to Weierstrass
Pretend you are about to participate in a debate, but you do not know yet which side of the question you will have to take in this debate. Thus, you must develop convincing arguments on both sides. Here is the question.
Resolved: The main ideas of calculus were completed by the time of Euler. For all intents and purposes, nothing that happened after 1790 makes any difference to the subject, except possibly to a tiny cult of mathematicians.
Write about a half of a page, or more, on each side of this question - that it is true and that it is false. Make the strongest argument you can on each side. Keep your arguments on the two sides separate. Be as specific as you can. Simply repeating pieties about mathematics being a rigorous subject will not suffice. Notice that those who agree with the above statement would strongly support elimination of the concept of limit from calculus, except in the most informal sense in which Newton would have understood it.
Don't forget Assignment 4 also includes a problem in Chapter 2.
Reflection on the Optimization Principle
On Wednesday (1/26) in class, we discussed how Mr. L's remark on p. 84, "It's all about when the two times are changing at the same rate. That's when the days are the shortest." The book further discusses this "Optimization Principle" on pp. 87-88. For this assignment, write a paper about 2 pages long (as usual, single-spaced, with an extra line between paragraphs) about this principle.
State clearly what the principle says, and how and when you can apply it. You should explain why it is true in terms of graphs and illustrate in several different contexts (bank account, reservoir, cars as in Vignette #1, etc.) Assume that your reader knows some calculus and knows a bit about these contexts, but doesn't understand what's behind the Optimization Principle. (It wouldn't hurt to remind your reader that the tangent line is horizontal at low and high points on a graph, if you use that idea.) End your paper by coming back to the "length of day" context, and explain how it can be that on the shortest day of the year the sunset has been getting later for a week or so, and yet, the sunrise has not yet reached its latest time.
Comments on classwork, possibly helpful when you're working on this paper. Be careful when you say "graph" that it is clear which graph you mean. In your paper, I'd like to see a few more steps of reasoning than in the classwork. Assume your reader doesn't get it if you just say, "When the tangent lines are parallel, the distance between the graphs is shortest." It may be helpful to discuss what it means when the lines aren't parallel. (Also you should be careful about the logic. In the context of Fig. 15, parallel tangent lines implies shortest distance between graphs. For your statement of the principle, is this the best choice for the direction of the implication?) I summarized the whole class discussion by saying, "The difference of the slopes of two graphs is the slope of the difference between the two graphs." If you use this idea in your paper, you should assume that summary, by itself, sounds magical to your reader, and needs a lot of clarification.
Don't forget Assignment 7 also includes questions in Chapter 3.
Question on SR and ACC*
Generalize your work on parts (c) and (d) in the §3.4 Exercise from class to a sequence of arbitrary length. That is, start with a sequence {x0,x1, ... ,xN}. (The 0, 1, and N are supposed to be subscripts, though in my browser they don't look very "sub"!) For each part, show at least one intermediate step, so your reader can see the result of applying each of the two operations. In all work and answers, write sequences out the way I did the initial sequence above: show at least initial and final entries. Do not use the "...i=0N" shorthand.
After you have done the generalization of the exercise, compare your work to the Proof of the Round-Trip Theorem on pp. 190-191. The book's proof depends on previous results from pp. 127-128. So it emphasizes the relationship to previous work; but to understand it, you have to remember well, or go back and study again, the previous work. The proof you just did, by generalizing the exercise, is a more direct, "bare hands," proof. What are the advantages and disadvantages of these two approaches? Which speaks more clearly to you? Why might the other method be more helpful to some students, or for some purposes? Write at least a paragraph (i.e., several sentences) addressing these issues.
Revised version of Q6 in §4.4 (p. 228)
For each graph that is given, draw the Asf graph. Then draw the possibilities for the original graph. We discussed in class how to to this for the first graph given, F '', and from the ideas suggested in class I have distilled the following requirements. The derived graph F ' = Asf(F '') + c. Draw examples of the original graphs for at least three possibilities: 0 < c, -60 < c < 0, and c < -60. Include the omitted cases, c = 0 and -60 either as additional graphs, or by changing some of the "less than" signs to "less than or equal to" signs in the three ranges listed above. In the latter case, describe briefly how the graph is different at these special values. If you have time (I know it's the end of the quarter) and want to impress me with the thoroughness of your answer, discuss how the graph of F varies as c varies within each range.
Handle the other two parts of the problem (G '' and K '') similarly.
Two of you drew lots of graphs for this problem the first time. So you may resubmit your previous version, possibly adding a few comments to fill in details I asked for above.
Comments on §5.5
Important point I forgot to note in class: In §5.5, we deal only with Symbolic Accumulation, corresponding to the ACC operation for sequences. The computations that correspond to the ACC* operation are in §5.6. The Round Trip Theorem related to §5.5 uses only the Successive Difference Operation (the numerator in the problems you've done for §5.4), NOT the Successive Rate Operation (the difference quotient).
The examples in the section all use functions that are increasing, and your homework problem involves a decreasing function. So there could be some ambiguity about the beginning and ending terms in your sums. Notice that the bar graph pictures show bars that meet the graph at the top LEFT corner of the bar. Use that rule in the homework question. So your accumulation sum for a function f(x) from 0 to p = mh should start with f(0) and end with f((m-1)h).
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Most recently updated on March 3, 2011.