Cartoon Calculus: Ch 4 (The fundamental theorem)

Quick links to Front matter, Back matter, and:
Part One: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits
Part Two: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics
Part Three: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

WM: I like the fact that you mention the fundamental theorem of calculus almost at the very beginning of the book. YB: Thanks! (There’s also a brief mention in the very first chapter :)

Page 43: Whee!

Pages 44-45: So far we’ve learned that derivatives… // Integrals involve adding up things…

MT: Page 44: clear explanation of derivatives. YB: Thanks!

WM: Page 44 – you state that we’ve learned what “derivatives” are, but unless I missed it, you haven’t used that word in the book yet. You’ve said that we solve these problems using “differential calculus”, but a beginner won’t know the relationship between the two words. (I expect this is a side effect of writing the chapters out of order.) YB: We used “derivatives” on page 4, but you’re right that that comes pretty early in the book. Maybe it would be better to edit p28 to say “derivatives, which is the subject of differential calculus” instead of just “differential calculus”? I’ve suggested that on p28.

RS: Chapter 4 opens with a definition of derviatives of “taking something really small and dividing it by something else really small.” That’s not a fair definition to me, because “really small” is relative. I think a better way to define derivatives in this context is is that “derivatives are about dividing something up into smaller pieces to understand the overall trend.”… The text below, I feel, should reference the common understanding of speed. In my experience students have a hard time with the formal definition but they have heard “miles per hour” a zillion times and “sixty miles travelled in one hour” is not a big jump. So even though this was covered in earlier chapters, I think a better way would be: “Speed is distance versus time. As in miles per hour!…So let’s take the distance you just traveled and divide it by the amount of time that just elapsed…and that’s deriving your speed.”… YB: We’re trying to include something about MPH on this page, but I’m not sure it will fit.

On page 45, I also don’t like the “really small” reference. I want to see something like “Integrals are combining a bunch of pieces together to get the entire thing. That’s addition, or if they are all the same—multiplication!” That way you can connect the division/multiplication reference at the bottom of the page. YB: On the one hand, I agree that this material isn’t perfect: it’s a bit of a stretch, but the point is to make a division/multiplication analogy for the Fundamental Theorem. My sense is that you don’t like the analogy and so the stretch doesn’t seem worth it to you. But IMHO the stretch is worth it. It’s a judgment call I guess. RS: I’m okay with the idea that lots of things in mathematics are the opposite of one another. Operations tend to come in complements (addition vs subtraction, multiplication vs division, exponents vs logarithms, etc.) But integration isn’t “super” multiplication. If it’s anything, it’s “super” addition. YB: Agreed. But I’m reminded of a line about the value of education, maybe it was something from a Bulwer-Lyton Bad Writing Contest from years past: “education is like a warm oven: the food is gone, but the heat remains.” You keep focusing on the shortcomings of the food, but I’m focused on the heat! Perhaps there’s a way forward that involves changing the following line in the book: “So if derivatives are like division, and integrals are like multiplication, then maybe they’re connected!” What you’re saying is that integrals aren’t actually much like multiplication, and for that matter it’s also true that derivatives aren’t really much like division, and what I’m saying is that I need a bridge to get to the connection between them; what I want readers to come away from this chapter with is not the idea that integrals are like multiplication or that derivatives are like division but that integrals and derivatives are connected. [RS: Yes! I don’t think you need to make the explicit connection to integrals/multiplication and derivatives/division, but the notion of compliments or opposites.] So maybe the text would be better as “So if derivatives have a conceptual connection to division, and integrals sometimes have a conceptual connection to multiplication, then maybe they’re conceptually connected to each other!” The challenge here is that this is too complicated and is way too many words. But I’m open to suggestions if you think this could be a way forward! PS. I’ve suggested to Grady the following language: So if DERIVATIVES are kind of like DIVISION and INTEGRALS are sometimes like MULTIPLICATION, then maybe they’re CONNECTED TO EACH OTHER!

MT: Page 45: change “sliver” to “slice” when describing the pizza. We don’t think it’s a common word to describe a slice of pizza and therefore could potentially stump or confuse a student, diverting his/her attention away from the explanation of integrals. YB: I have suggested this to Grady. Note that “sliver” is used in the previous chapter (pages 37 and 40).

Pages 46-47: Now, we all know that… // It turns out that…

MT: Page 46: not sure if this is already being done, but it would make it clearer if the character who is thinking of a number could have a thought bubble above his/her head with the number he/she is thinking about (example is attached). YB: I have suggested this to Grady, but I’m not sure if there’s space to do this (especially on the next page, as you suggest below) and I’m also not sure this is a good idea because the point is that reader, like the first character, can figure out what number the second character is thinking of.

MT: Page 47: Include a thought bubble with the function that he/she is thinking about. We also felt like it would be helpful and add clarity if an equation were included under “holy cow, that’s the fundamental theorem of calculus.” YB: Discussed above.

WM: Pages 46-47 are very clever. YB: Thanks!

RS: The small text on Page 46 to plan to make both of these characters female irked me, but that might just be me. Hard to tell in context. YB: We’re just struggling to deal with the gender imbalance in the book. If you know of any women mathematicians who made noteworthy contributions to calculus then please let me know! RS: The name that comes to mind is Sofia Kovalevskaya! YB: Good suggestion… we’ll try to find room for a mention in the book. But where? I suppose we could have her name somewhere on this page, but if there’s a better place for it that would be swell. Update: I’ve asked Grady for his thoughts on including Sofia somewhere.

RS: On page 47 the parallel joke is cute but I don’t know that people necessarily understand functions yet. YB: Yeah, figuring out how much background info to give to (or assume from) readers is a challenge. Larry Gonick has a Cartoon Guide to Calculus and he spends the first 50 pages on Chapter 0, which is all about functions etc. Our approach is basically to assume that readers know something about functions, or that they’re willing to learn it on their own. RS: I don’t think you need a lot of formal definition here: a function is something which has an input and an output and behaves in a predictable way. You could do it one page, with a little machine that doubles numbers, one that folds paper into airplanes, or one that incenerates garbage. YB: I think we’ll tackle this in the glossary. [RS: I don’t know that it’s worth doing if you don’t include it right up front. I actually like the idea of a couple of one page “you must know this before reading this book” stuff like how to read an X/Y graph, what a function is, what an equation is, and the notion of variables. But I’m really adding a lot to that.] My concern here is partly about interrupting the flow of the narrative, and partly about half-assing a definition. I think a definition should at least cover the idea that for each input value there is at most one output value, and that’s a lot to chew on because it brings up issues of domain and such. Plus there’s the following consideration: If you’re somebody who doesn’t have an intuitive understanding of functions, then are you really going to be reading this book? [RS: I think the answer is yes. There’s a great book by John Bruer called Schools For Thought which covers many of the findings about mathematics education, especially that people memorize cookbook-y patterns and sequences and mostly make errors in following the recipe. I think a big part of your audience is people who never really understood what was happening in algebra class but managed to pass.] And if you are then how much of the Fundamental Theorem are you really going to understand anyway? And how much value-added can this book provide in boosting that understanding? The risk here is the same risk that I sometimes encounter with economics comedy: “explaining the joke” sometimes ruins the joke or throws off the timing to such an extent that it’s better to just charge ahead.

RS: For me the big advantage of a book like this is that it can give readers a conceptual framework for calculus without having to learn a lot of arithmetic or symbolic manipulation. To me a tragedy of the school system is that kids can still recite the quadratic formula decades later, but they don’t see how everyday phenomeona could be described as a polynomial. I don’t particlarly need for readers to be able to do integration by parts. But I would like for them to be able to see problems in the real world and say “that’s a calculus problem!” or hear about derivatives in the financial news and have a sense that people are betting on rates of change instead. YB: I think we totally agree about this big-picture goal. The challenge is how to get there :)

Pages 48-49: To continue the analogy… // It turns out…

MT: Page 48: Good! Simple and clearly explained and illustrated. YB: Thanks!

ME: “to continue the analogy, imagine that division was were really difficult” subjunctive form of to be YB: Hmm… that sounds stilted to me, but I will ask our editor.

RS: I’m not a fan of the idea that notion that derivatives are easy and integrals are hard. This is a property of our notation and our training more than anything innate to mathematics. I don’t have a good alternative yet. YB: Hm… This is probably going to stay as-is but you make an interesting claim. It’s definitely true that we pretty much only use the Fundamental Theorem to solve integrals (by turning them into reverse derivatives, or however you want to think about it) but is that because of notation and training or because of something intrinsic in the mathematics? I’m tempted to say it’s the latter, but I haven’t thought about it deeply. RS: I suppose in the future when you’re writing the “Cartoon Guide to Fourier and Laplace Transforms”, we can have a conversation about why it’s “easier” to solve certain kinds of equations by moving them out of the time domain into the frequency domain. What I want readers to understand is that there is an intrisic mathematical discovery here that happens in layers. First, we describe fixed relationships as functions. And then we can characterize the behavior of those functions as a derivative, which is another function. But what does it means to study a derivative and try to understand the original function it characterizes? That’s what this is about, in my view. YB: I must confess that I went off into number theory before getting to Fourier and Laplace transforms, so the cartoon book you suggest will have to wait for a better-qualified author! As for your description of “what this is about”, I guess I’ll repeat that you make an interesting claim and I’ll confess that I’m not sure what to do with it. To pick just one challenge, there are of course a whole class of “original” functions that are characterized by the same derivative. You and I understand that, but for someone who’s just starting out in mathematics that gets pretty complicated IMHO. [RS: To me this is a great example of a how a little intutive thinking can help people get a calculus mindset. Bob is driving along the highway and passes Smithville. Twenty minutes later, Bob passes Amitytown. Mary starts driving on the same road two hours later and passes Smithville, and then twenty minutes after that, she passes Amitytown. Based on what you know, who is driving faster---Bob or Mary? Or are they traveling the same speed? That is a calculus problem where there are different functions (Bob’s location and Mary’s location with respect to time) but they have the same derivative.]

RC: Ziplines: This is one of many references to ziplines, this one talks about “a zipline that helps us avoid the difficult parts” by “climbing mount derivative.” Ziplines use gravity to let people to descend from high to low as in sketch on first page of chapter, except this p 49 sketch has the person going low to high. (also mentioned on pg 134, and the same joke is on 147 but the sketch makes sense)… Rappelling (abseiling) also gets you down fast — so does glissading (on scree or ice)… For going up fast, cable-cars (gondolas) are great – we use them to get up high for multi-day hikes and climbs, also funiculars, téléphériques, skyways, aerial lifts. YB: This is interesting info, but I think we just need more clarity with the graphics and text on p49. The zipline on this page is going high to low.

MT: Page 49: This page really stumped us. On page 45 division is likened to derivatives and on page 47 division is described as “really difficult,” but on page 48 it says, “calculating derivatives is easy.” We aren’t sure if readers are going to look into the text as deeply as we did, but we nevertheless noticed the inconsistency. YB: Good catch, but Yes you’re reading into it a little too deeply! There is an inconsistency here, though, and I’ve suggested to Grady some text changes that might help soften the rough edge here.

Pages 50-51: The math of the fundamental theorem… // There’s even an easy way…

MT: Page 50: Great! YB: Thanks!

MB: on p. 50 you say derivatives and integrals are ‘opposites’. They are, rather, inverses of each other: any person who has had a class in high school math will have heard of inverse functions, and the idea that one operation inverts another is attractive. YB: Good point, but it’s a question of whether the greater precision in language is worth the confusion it might cause for folks who don’t know what inverses are. I’m inclined to leave it as-is but will discuss with Grady.

MT: Page 51: We think “easy” should be changed to “easier,” since on page 50 the meaning of the fundamental theorem is described as “simple.” After the change, it would be: “There’s even an easier way to understand the intuition behind the fundamental theorem.” Also–it’s great that a reference to “the formal definition of a function” in the glossary is made! YB: I think “easy” is fine as-is; it’s about the intuition behind the Fundamental Theorem, whereas on p50 what “simple” is the meaning of the Fundamental Theorem.

WM: Page 51 – “squiggly line on a graph” might be confusing. The Squiggly line IS the function, but a reader might think you mean the axes are the function. (I’m not sure I’m being clear here about where I think the confusion lies, but my solution is to say “squiggly line on a coordinate axis”.) YB: Maybe it’s better if we take out “ON A GRAPH” and just say “squiggly line”? In any case, “coordinate axis” is too much of a mouthful.

BG: do people need a reminder of what a function is? The math text trope of the black box where you put in x and out pops f(x) which also happens to be the y value on the graph? YB: Yes, and that’s why we’re referencing the glossary.

Pages 52-53: We can use the integral… // It’s pretty obvious that…

MT: Page 52: Good. YB: Thanks!

WM: Page 52 – have you explicitly stated that an integral calculates the area under a curve before this page, or just implied it? YB: I think it’s explicit on p31 and p41.

WM: I don’t quite follow your argument on page 53. YB: This is an attempt at an intuitive explanation of the Second Fundamental Theorem. I’m afraid I’m not sure how to do it better.

MT: Page 53: Nice summary at the bottom of the page. YB: Thanks!

Page 54: It’s a little harder…

MT: Page 54: This applies not just to this page, but we like how the statement “derivatives and integrals are opposites” is repeated numerous times throughout the chapter; this makes it very clear to the reader what the main point is. YB: Thanks!

Quick links to Front matter, Back matter, and:
Part One: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits
Part Two: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics
Part Three: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

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