MA177--How to prepare for the final exam

To help you prepare for the final, here is a list of things you should review. I've tried to cover most of the important things we discussed, and I've also written a few sample essay questions. The exam will have a few short answer questions about concepts and/or people, followed by a couple of essay questions. Feel free to come talk to me at any point if you need help with any of this!

Good luck!

Concepts: You should be able to give a short description of each of the following.

The Königsberg bridge problem Scientific academies The Ladies' Diary Newton's Principia Newton's Opticks Universal gravitation Newtonian mechanics Newtonianism on the Continent The problem of the Earth's shape Maupertuis' expedition to Lapland The problem of the vibrating string Leibniz's "principle of continuity" The principle of least action Laplace's Mécanique Céleste Lagrange's Mécanique Analytique "Social choice theory" (elections) "Metaphysics of the Calculus" Fluxion Differential Derivative

Berkeley's critique of the Calculus D'Alembert's concept of "limit" The "problem of the points" The Fermat-Pascal letters Probability Expectation Jakob Bernoulli's Ars Conjectandi The St. Petersburg paradox Smallpox inoculation "Probability of judgments" Probabilistic/traditional insurance Number theory Perfect numbers Mersenne primes Pell's equation Fermat's "Little Theorem" Fermat's "Last Theorem" The four squares theorem Primality testing

People: You should be able to describe the mathematical and scientific ideas and/or contributions of the following people. (I don't mean dates and other biographical facts, but you should know some general information about their ideas, their "style," their importance, and their influence.)

René Descartes Isaac Newton Gottfried Leibniz Pierre de Fermat Blaise Pascal Jakob Bernoulli Voltaire Émilie du Châtelet Pierre de Maupertuis Alexis Clairaut

Leonhard Euler Daniel Bernoulli Denis Diderot Jean-Jacques Rousseau Jean Le Rond D'Alembert Pierre Simon de Laplace Marquis de Condorcet Adrien-Marie Legendre Joseph-Louis Lagrange

Sample Questions: Here are some sample essay questions. You should expect to see similar ones on the exam.

• How did Newtonian ideas about gravitation and about mechanics become accepted in continental Europe? To what extent did this acceptance depend on the correctness of these ideas?

• How did Newton explain the "action at a distance" (e.g., the Sun exerts an attractive force on the Earth despite the fact that the Earth is very far away) that his gravitational theory implies? How did other scientists think about this problem?

• Maupertuis' "principle of least action" said that nature always acted in such a way as to minimize a computable quantity called the "action." The principle is essentially correct. Maupertuis claimed that this principle could be viewed as evidence for the existence of God. Voltaire thought this was very funny. What do you think?

• Explain the importance of the idea of the "reasonable man" in the early theory of probability. What caused this idea to become problematic?

• What does studying the history of the application of probability theory to various different real-world problems tell us about how mathematics gets to be applied to the world?

• Explain Newton's notion of "the fluxion of a flowing quantity" and Leibniz's idea of "the differential of a quantity." How are they related?

• What was the goal of Berkeley's The Analyst? What were the crucial points in his critique of the Calculus? How did mathematicians react to Berkeley's critique?

• Explain what Condorcet discovered about elections and how it might impact the democratic process.

• Discuss Judith Grabiner's summary of the history of the derivative: "first used, then discovered, then developed and applied, and finally defined."

• Describe the "problems concerning numbers" which Fermat studied and tried to get other mathematicians to study. Speculate about why he had so little success in capturing the interest of others.

• Assess Euler's role in science and mathematics in the eighteenth century. What sorts of things did he do? Was he influential?

• Some of the great mathematicians of the eighteenth century were deeply involved in the program of the "philosophes" and are well-known Enlightenment figures. Others focused almost exclusively on science and mathematics. Give examples of each kind, and speculate on the reasons for the difference.

• State and prove Fermat's "Little Theorem."

• Is there a difference between eighteenth-century "mixed mathematics" and modern "applied mathematics?" If so, what is it? If not, why the change in terminology?

• What was Jakob Bernoulli's crucial difficulty when he wanted to move probability theory from a focus on games of chance to more "serious" applications? How did he attempt to solve it?

• Explain the St. Petersburg paradox, and discuss the various explanations and resolutions of the paradox that were proposed in the eighteenth century.

• "At the end of the seventeenth century, the idea that one should use mathematics to understand the world was still somewhat controversial. At the end of the eighteenth century, it was a given." Do you agree? If so, what caused such a fundamental change in attitude? If not, explain why.

• In the nineteenth century, more and more mathematicians focused their work on "pure mathematics," ignoring applications. Is there anything in eighteenth century math and science that might suggest this development?

• Both Diderot and Rousseau felt that using mathematics to understand the world was a mistake. Why did they think so? Were their reasons the same? How did their point of view affect the attitude of the revolutionary governments towards the sciences?

• In the first half of the twentieth century, the British Navy used the following method to test shells. The shell makers sent the Navy shells in lots of 400, subdivided into sub-lots of 100. When a lot was received, two shells were picked at random from sub-lot number 1. If the first shell was fired and worked correctly, the other 399 were accepted. If the first shell failed, the second was tested. If it worked, the remaining 398 were accepted. If both failed, the whole sub-lot was rejected, and the testers went on to sub-lot number 2, using the same procedure. Suppose that 50% of the shells are duds. What is the probability that either 399 or 398 shells from that lot are accepted?

• Explain how one can use "Fermat's Little Theorem" to test whether a number is prime. Is this an efficient method to test for primality? What are its pros and cons?

• Why was Alexis Clairaut's correct prediction of the exact date of the return of Halley's Comet important?

• Is Number Theory interesting? Why or why not?

• In a recent editorial in FOCUS, mathematician Keith Devlin argues that we should abandon the emphasis on teaching mathematical technique in middle and high school, and instead focus on teaching students about the history, uses, and nature of mathematics. He argues that for most students this knowledge is more useful than knowing how to do algebraic manipulations, and that the remaining students might be motivated by this kind of introduction to put in the effort to learn the technical stuff. In the light of what you've learned this semester, do you agree or disagree? (Generalities won't cut it in answering this question; be specific!)

• Why was probability theory such an important part of eighteenth century "mixed mathematics"? How successful was it?

• What was "social mathematics"?

• How did the "scientific revolution" in Europe affect life in America? Was there any significant science in America during the eighteenth century?

• During the eighteenth century, people like Voltaire, D'Alembert, Diderot, and Condorcet managed to do significant work in "scientific" fields, in what we'd now call "social science," and in the humanities. Nowadays such "crossover" is very rare. Why do you think this is? What are the crucial differences that lead to this split between the "two cultures"?

• What justifies studying mathematics? (Claiming that such study is not justified is a valid answer if you can give a good argument in support of your claim. In particular, discuss whether the case for or against mathematics would apply to poetry, music, biology, and economics.)

• Hans Freudenthal once wrote a paper called "Should a teacher of mathematics know something about the history of mathematics?" Since you've been learning mathematics for over twelve years, you actually have some expertise on the subject of how it should be taught. How would you answer Freudenthal's question?