A Tour of the Calculus Discussion Questions

General Questions

1. How does the author surprise you?
2. How does the author delight you?
3. How does he frustrate you?
4. What do you agree with?
5. What do you disagree with?
6. What questions arise as you read?

Introduction

1. What "division of experience" does Berlinski mention? Describe it.
2. How is mathematics related to the real world?

1. What is the goal of the book?

The Frame of the Book

1. Of what eight elements is calculus comprised?
2. Elaborate on each of these elements as we meet them in the book.

Chapter 1

1. What century contains the explosion of science of which calculus is a part?
2. Who was most instrumental in developing the calculus at the beginning?
3. Of the two large personalities mentioned, how did their visions differ? How did their personalities differ?

Chapter 2

1. How does elementary Euclidean geometry represent the world?
2. How does it differ from the world?
3. Leo Kronecker said that the natural numbers are god-given; which numbers are "natural"?
4. Beyond the natural numbers, what numbers were invented? How did they solve "deficiencies" in the natural numbers?
5. What is the difference between "how many" and "how much"?

Chapter 3

1. How are geometry (shape and space) and arithmetic (numbers) different from each other? Can you see the perspective that they describe completely different worlds?
2. What is the connection between geometry and arithmetic? What is the name of the field that draws this connection?
3. What part does the sign of a number (positive, negative, or zero) play in geometry?

Chapter 4

1. For whom is the Cartesian coordinate system named? What part does he play in history? in mathematics?
2. What does the author say about a line that is not embedded in a Cartesian coordinate system?
3. What does the coordinate system add to the line?
4. What grammatical metaphor does the author bring to the equation y = mx + b?

Chapters 5 – 6

1. What important number is Chapter 5 about?
2. What important theorem tells us this number should exist?
3. Why did the Greeks have problems with this number?
4. What does the taxi driver prove in Chapter 6?
5. Summarize his argument: write a sequence of equations, and then explain the contradiction.

Chapter 7

1. Paraphrase three ways of looking at a line that you find particularly appealing.

Chapter 8

1. To what "numbers" is the author referring when he says the line is a richer object than the numbers?
2. How would you describe the meaning of the statement, "The real line is a continuous object"?
3. How did Dedekind describe continuity?
4. Define the word converse of a statement. As an example, write the converse of the statement, "If it is raining, then I am getting wet."
5. How long did it take, after Newton and Leibniz, to build the logical foundation of the calculus?

Chapter 9

1. What is the difference between rational and irrational numbers in their decimal representation?
2. Give an example of a fraction p/q whose decimal expansion ends.
3. Give an example of a fraction p/q whose decimal expansion repeats but does not end.
4. What fraction, in form p/q, is represented by the decimal 0.12121212...?
5. The xy-plane can represent a 2-dimensional, static region in space. What new perspective allows the xy-plane to represent a changing, dynamic world?
6. If you don't like fractions and find the irrational numbers just too mysterious, what mathematician would you be drawn to?

Chapter 10

1. Compare and contrast the physician, the novelist, and the mathematician.
2. What does the squaring function do?
3. How is the squaring function defined?
4. What is the squaring function?

Chapter 11

1. How many grains of wheat did the caliph agree to supply? Can he do so? Explain.
2. What function families are most familiar?
3. What number is the "black jewel of calculus"?
4. After reading this section, for whom do you think this number is named?
5. What is the "most beautiful formula in all of mathematics"? (You will prove this formula in Calculus II.)

Chapters 12 – 13

1. Paraphrase the two questions posed by the student Inglefinger.
2. How would you answer those questions?
3. How does the author argue that the Cartesian coordinate system is "realer than real"? Do you agree? Explain.
4. What is Galileo's contribution to the development of calculus, and when did it occur?
5. Describe three different ways that you can average 70 miles an hour.
6. What is the formula for speed?
7. What is the problem in trying to assign a speed to one instant in time? Why do we want to?
8. Who first used the symbols dy/dt? Why those symbols, and what do they signify?
9. Who is Bishop Berkeley, and what did he contribute to calculus?

Chapter 14

1. According to the author, who is the first "modern mathematician"? How so?
2. What important deeds is he credited with?
3. Compare Leibnitz' definition of instantaneous speed (p.111) with the new definition in this chapter.
4. On p.124 the author refers to mathematics as a performing art. Explain, and provide another similar example.
5. The Appendix provides the definition of the limit of a sequence of numbers, something you will encounter in Calculus II. See if you can use the definition to provide a proof that the limit of the sequence , as n → ∞, is 0.

Chapter 15

1. Near the beginning of the chapter, the author states what he calls the "most compelling impulse" of calculus. What is that impulse, and why must calculus address it?
2. Explain the author's argument that f(x) = x2 is continuous at x = 3.
3. Explain the author's argument that f(x) = is discontinuous at x = 1.
4. How does the author use his walk through Prague to illustrate three major properties of continuous functions? What are they?

Chapter 19 – final exam question

1. Write, in your own words, a summary of this chapter, not including the appendix. Include important mathematical ideas, people and history, the view of the author, and your own opinions. Relate this chapter to the rest of the book, and to your studies of calculus. Turn this in at the final exam, to be worth 5% of your exam grade. It should be between two and three typed pages, double-spaced.

Chapter 20

1. Who is Igor M, and what is his story?
2. Give an example from everyday life of "versal and reversal": some process or activity that is undone by another.
3. What is the purpose of an equation that contains an unknown? If the unknown is a number, how is the equation structured? If the unknown is a function, how is the equation structured?
4. What is meant by differentiation being puritanical and antidifferentiation being promiscuous? How promiscuous is antidifferentiation?
5. What important principle appears in the section "And Yet Again the Mean Value Theorem"? How does the MVT figure in this principle? (You may need to read a few pages further to see what the author is getting at.)
6. How does calculus invest a simple physical fact with a complete description of freely falling bodies any time, anywhere?
7. After reading this section and the Appendix, what do you think of the author's class and mode of teaching?

Chapter 21

1. What are the two main concepts of calculus? How are they alike? How are they different?
2. On p.248, writing about imprecise terms such as area and territory, the author finishes, "... the conceptual work of the calculus as much a matter of clarification as discovery or definition." What does he mean? Give another example from our studies this semester.
3. What child's toy illustrates how to approximate the area of the region under a curve?

Chapter 22

1. How is simple Euclidean geometry connected to calculus? Where do they part ways?
2. Does the definite integral calculate area or define it?
3. Explain the content of the theorem described on pp.258-260. Where do you find the corresponding theorem in our textbook?
4. Two great mathematicians brought sense to the definition of the definite integral. Who were they and what did they contribute?

Chapter 24

1. How is the derivative both a number and a function?
2. How is the integral both a number and a function?
3. How is a football game similar to treating the integral as a function?
4. What is the difference between the definite integral and the indefinite integral?
5. What abuse of notation is the author guilty of?
6. Did you find the typo at the bottom of p.279?
7. How does the number e arise in integration?
8. Can "any idiot" learn math by practicing patience?

Chapter 25

1. How do both math and art unite the living and the dead?
2. On what stage does the FTOC perform? Who are the actors in constructing the definite integral?
3. What aspects of the integral are living, and which are dead?
4. The author states that the two parts of calculus involve, respectively, contraction and expansion. Explain.
5. What is animal empiricism? How does it relate to calculus?
6. Who is Isaac Barrow?
7. In what units is given, ft or ft2? Explain.

Chapter 26

1. What is the author's view of the possibility of eventually achieving a "final theory" of the physical world? What is yours? Does each advance bring us closer?
2. According to the author, what is the difference in spirit between physics and molecular biology?
3. Which do you find more appealing, the approach of mathematics that goes for depth and subordinates reality to theories, or the approach of biology that goes for breadth and welcomes the "clutter of experience"?
4. Whither continuity, according to the author?

More General Questions

1. What is the historical outline of the invention of calculus? Start with 200-400 BC and end ... where?
2. Who are the major players? List at least five.
3. What is calculus? Elaborate, but confine your answer to one hand-written page.
4. Would you recommend A Tour of the Calculus to a friend studying calculus or science? Why or why not?