Cartoon Calculus page notes

Below please find page notes for The Cartoon Introduction to Calculus. This is a working document, so additions or edits are welcome! Also note that occasional Wikipedia references are for topics that can be found in many introductory textbooks.

Chapter 1: Introduction (pages 3-14)

Page 7, Newton and Leibniz: Isaac Newton (1642-1726) and Gottfried Wilhelm Leibniz (1646-1716) share credit for inventing calculus. For more on the feud between them, read The Calculus Wars by Bardi.

Page 8, “Mt Fluxion”: Newton used terms like “fluxions” and “fluents” in his papers on calculus. These terms are now historical curiosities.

Page 8, Liu Hui and Archimedes: Liu Hui was a 3rd-century Chinese mathematician. Archimedes was a Greek mathematician and scientist who lived a few hundred years earlier. They independently worked on ideas that relate to calculus.

PS. The Chinese characters on the rock (“路漫漫其修远兮,吾将上下而求索”) are from the poet Qu Yuan, who lived about the same time as Archimedes (3rd century BCE). The Google translation is “The road is long and the road is long, I will go up and down.” The translation from a friend is “there is a long way to go to solve the problem, I will continue searching to figure it out”.

Page 11, “Let’s define zero as the empty set”: Set theory is a branch of mathematics that was used to try to provide a solid foundation for the rest of mathematics; in my undergrad days I read a great book about it, maybe Naive Set Theory by Halmos. Another book to try is The Joy of Sets (!) by Devlin. My recollection is of a field of study that was austere and kind of fascinating: define “zero” to be the empty set {}, “one” to be the set containing zero, i.e., {{}}, “two” to be the set containing zero and one, i.e., {{}, {{}}}, and then you can define operations in such a way that one plus one equals two! Maybe not for everyone, but IMHO really cool.

Chapter 2: Speed (pages 15-28)

Page 15, “60 miles per hour”: This treatment was inspired in part by the Feynmann Lectures on Physics (read them for free here) and in part by calculus notes from my undergraduate professors at Reed College: here are Ray Mayer calculus notes and Jerry Shurman calculus notes.

Page 21, “fluxions”: See page 8.

Page 21, “ghosts of departed quantities”: The philosopher George Berkeley (1685-1753) objected to the approach of calculus in a book published in 1734 called The Analyst. (Full name: The Analyst: The A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.) In the book he mocked the ideas of differential calculus (“And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”) and in so doing helped promote research that put calculus on a firmer foundation. For more on all this see the lecture by Raymond Flood on “Ghosts of Departed Quantities: Calculus and its Limits”.

PS. Berkeley is perhaps most famous for his philosophical beliefs—my limited understanding is that he believed that a tree only existed if somebody was there to observe it—that apparently were the subject of two limericks by Ronald Knox:

There was a young man who said “God
Must find it exceedingly odd
To think that the tree
Should continue to be
When there’s no one about in the quad.”

“Dear Sir: Your astonishment’s odd;
I am always about in the quad.
And that’s why the tree
Will continue to be
Since observed by, Yours faithfully, God.”

Page 23, Newton’s apple: The idea that Newton came up with the theory of gravity by watching a falling apple is apparently true! See “Newton’s Apple: The Real Story” in New Scientist, with more on Wikipedia.

Chapter 3: Area (pages 29-42)

Page 33, Cavalieri: Bonaventura Cavalieri (1598-1647) was an Italian mathematician whose ideas about “indivisibles” foreshadowed integral calculus.

Page 34, Liu Hui and Archimedes: See page 8. Note that the latest estimate of pi (as of March 2019, from Emma Hiruka Iwao) goes out more than 31 trillion digits.

Page 34, “It’s irrational!” This is a (time-worn) joke about pi being an irrational number, i.e., one that cannot be expressed as a fraction of whole numbers. Here are more jokes about pi.

Page 38, “How did they do it?”: The general approach here is called the method of exhaustion.

Page 39, Pythagorean Theorem: One of the most famous theorems in mathematics. Despite the great quote from the basketball legend Shaquille O’Neal (who once said that his game was “like the Pythagorean Theorem: hard to figure out”) there are in fact hundreds of proofs of the Pythagorean Theorem going back thousands of years.

Page 39, “repeat this over and over again”: Note that repeating the process is a bit tricky.

Page 40, estimate with 12,288 sides: See the work on Zhu Chongzhi.

Page 42, Sofia Kovalevsky, Isaac Newton, and turtles: The woman at the top of page is Sofia Kovalevskaya (1850-1891), called “the greatest known woman scientist before the twentieth century”. She was a Russian mathematician who advanced our understanding of calculus. The Newton quote about “standing on the shoulders of giants” goes back (appropriately enough) at least as far as the 12th century. The joke at the bottom (“It’s mathematicians all the way down”) is a reference to the cosmological idea of “turtles all the way down”.

Chapter 4: The Fundamental Theorem of Calculus (pages 43-54)

Page 50, Fundamental Theorem of Calculus: Just about everyone agrees that the Fundamental Theorem of Calculus has two chunks, but the universal agreement stops there. We call these two chunks Part 1 and Part 2 of the Fundamental Theorem of Calculus, but some texts call them the First Fundamental Theorem of Calculus and the Second Fundamental Theorem of Calculus. To complicate matters even further, some texts also flip the order, i.e., what we call Part 1 of the Fundamental Theorem is not only also sometimes called the First Fundamental Theorem but is also sometimes called Part 2 of the Fundamental Theorem, or the Second Fundamental Theorem. Although this can be confusing if you’re looking at multiple texts, the underlying mathematical statements are of course true regardless of what you call them.

PS. Note that there are other fundamental theorems in mathematics, e.g., the Fundamental Theorem of Arithmetic, which says that any number can be expressed in one and only one way as the product of prime numbers. Here are some other fundamental theorems in mathematics and beyond.

Chapter 5: Limits (pages 55-68)

Page 56, Zeno: A greek philosopher, Zeno (approx. 490-430 BCE) is best known for his paradoxes.

Page 64, epsilon-delta definition of limit: See wikipedia.

Chapter 6: Limits and Derivatives (pages 71-82)

Page 72, speed and velocity: Velocity is a vector, i.e., it has a direction. Speed is a scalar; more specifically, speed is the absolute value of velocity. (It could be that your velocity in a given direction is -30 mph, i.e., you’re going the other way; but your speed in this case is still 30 mph.)

Page 77, ways of writing derivatives: See wikipedia for more on these variations.

Chapter 7: The Calculus Toolkit (pages 83-94)

Page 91, product rule: Note that this “intuitive approach” actually hides something: take the original rectangle, add on the rectangle for the change in height and the rectangle for the change in width… and you don’t quite get the new rectangle because there’s a missing square in the upper-right-hand corner. This “missing square” turns out to disappear (because of limits!) when you do the actual math, but it was a topic of discussion for critics of calculus like George Berkeley. (See page 21 above and also discussions like the one starting on page 28 of this paper by Katz and Sherry: “Leibniz’s infinitesimals: their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond”).

Page 92, induction: More on wikipedia.

Chapter 8: Extreme Values (pages 95-106)

Page 97, “Eek! It’s a talking worm!”: A reference to the “talking muffin” joke about two muffins in the oven. The first says “Is it just me or is it getting hot in here?” And the second says “Eek! A talking muffin!”

Page 106, derivative is zero but it’s not an extreme value: For more on how to check this, see the second derivative test.

Chapter 9: Optimization (pages 107-118)

Chapter 10: Economics (pages 119-130)

Page 123, marginal analysis: For more, see our two-volume Cartoon Introduction to Economics.

Page 130, blah blah blah: Inspired by one of Gary Larson’s Far Side cartoons about “What we say to dogs” versus “What they hear”.

Chapter 11: Integration the Hard Way (pages 133-146)

Page 133, “a journey of a thousand miles begins with a single step”: A quote from the Chinese philosopher Lao Tzu (aka Laozi).

Page 136, Riemann sums: Named after Bernhard Riemann (1826-1866), a German mathematician. (Note that Riemann sums don’t always have to be evenly spaced.) Also named after him is the Riemann hypothesis, one of the most famous unsolved problems in mathematics.

Page 138, Kangshung Face: One of the routes up Mt Everest.

Page 139, Lebesgue measure: An extension of the methods of calculus to areas that cannot be evaluated using Riemann sums. Named after French mathematician Henri Lebesgue (1875-1941).

Page 142, infinitely many paths: This is a joke because infinity minus one is still infinity. But there are in fact different kinds of infinity, some larger than others. See the work of German mathematician Georg Cantor (1845-1918), and note in particular that a “countably infinite” set of numbers is one that can put in a one-to-one correspondence with natural numbers; for example, the set of even numbers is “countably infinite” because you can say that the 1st one is 2, the 2nd one is 4, the 3rd one is 6, and so on. In contrast, the set of real numbers is not countably infinite, it is in some meaningful sense “larger” than the set of natural numbers. To see this, use proof by contradiction: assume you can list all the real numbers as “the 1st one is x, the 2nd one is y, etc.” Now consider the number that differs from x in the first digit after the decimal point, that differs from y in the second digit after the decimal point, etc. This produces a real number that is different from all of the numbers in the list, so we have created a contradiction and can conclude that the set of real numbers is not countably infinite.

Chapter 12: Integration the Easy Way (pages 147-158)

Page 148, matter and anti-matter: More on wikipedia.

Page 150, good and bad anti-derivatives: A “bad” anti-derivative is one that basically uses the Fundamental Theorem of Calculus in an unhelpfully self-referential way.

Page 155, moonwalking: Here’s a video of how to moonwalk. Popularized by Michael Jackson (see the 45-second mark here, or the 3:40 mark here), the move actually goes at least as far back as the 1930s.

Page 157, Chekhov’s gun: The Russian playwright Anton Chekhov (1860-1904) gave the following famous advice, known as “Chekhov’s gun”: “Remove everything that has no relevance to the story. If you say in the first chapter that there is a rifle hanging on the wall, in the second or third chapter it absolutely must go off. If it’s not going to be fired, it shouldn’t be hanging there.” (See page 185.)

Chapter 13: The First Fundamental Theorem, Revisited (pages 159-172)

Page 160, “you can’t handle the truth”: A reference to the 1992 move A Few Good Men. See the clip here.

Page 171, “we’re too chicken to write it down”: This is a reference to one of the most famous problems in the history of mathematics, Fermat’s Last Theorem. The theorem refers to the idea that the Pythagorean Theorem (a^2 + b^2 = c^2 with integers such as a=3, b=4, c=5) cannot be extended to higher numbers, i.e., that there are no integer solutions to a^x + b^x = c^x for x>2. Pierre de Fermat (1607-1665) was a French lawyer and mathematician who made many notes in the margins of one of the math books in his library. One of those notes referenced this idea, saying “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”. Similar claims elsewhere in his notes were all eventually proved, but this “last” theorem defied countless attempts to prove it and became one of the most famous unsolved problems in mathematics… until Andrew Wiles proved it in 1994, more than 300 years later.

Page 172, “one ring really does rule them all”: A reference to The Lord of the Rings.

Chapter 14: Physics (pages 173-186)

Page 174, invade Egypt: See Wikipedia.

Page 174, Newton’s to-do list: According to Wikipedia: “In 1666 Isaac Newton, aged 23, made revolutionary inventions and discoveries in calculus, motion, optics and gravitation. As such, it was later called Isaac Newton’s Annus Mirabilis ["miraculous year"]. It was in this year that Isaac Newton was alleged to have observed an apple falling from a tree, and in which he in any case hit upon the law of universal gravitation (Newton’s apple). He was afforded the time to work on his theories due to the closure of Cambridge University by an outbreak of plague.”

Page 175, Newton’s laws of motion and gravitation: See NASA or this Q&A page from the University of Illinois.

Page 176, Newton/Leibniz argument: For more on the feud between them, read The Calculus Wars by Bardi.

Page 180, Major Tom: A reference to David Bowie’s song Space Oddity.

Page 185, Chekhov’s gun: See page 157.

Page 186, turn lead into gold: Newton devoted considerable energies to occult studies such as alchemy (e.g., trying to turn lead into gold) and attempting to determine from the Bible when the world would end. Wikipedia puts it this way: “After purchasing and studying Newton’s alchemical works, [economist John Maynard Keynes] opined in 1942… that ‘Newton was not the first of the age of reason, he was the last of the magicians.’”

Chapter 15: Limits Beyond Limits (pages 187-200)

Page 189, knot theory etc.: See Wikipedia on knot theory, and also these videos.

Page 190, fractals: See the Fractal Foundation.

Page 191, Ramanujan: The Indian mathematical savant Srinivasa Ramanujan (1887-1920) rose to the top of the mathematical world after he sent some of his mathematical discoveries to the famous British mathematicians Hardy and Littlewood (whose abilities and collaboration were such that it was said that the three most famous British mathematicians of the time were “Hardy, Littlewood, and Hardy-Littlewood”). Hardy quotes Littlewood as saying, of Ramanjuan, that “Every positive integer was one of his personal friends.” Another Hardy story:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No”, he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

For more, see the movie Ramanujan or the book or movie The Man Who Knew Infinity.

Page 191, central limit theorem: See Wikipedia for more on this key theorem in statistics.

Page 192, Euclid:

Page 193, Prime Number Theorem:

Page 197, differential equations, population dynamics:

Page 198, multi-variable calculus:

Page 199, last dollar rule:

Glossary (pages 201-207)

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