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…

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 and differential calculus” instead of just “differential calculus”?

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.”… 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.

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

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: 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.

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

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.

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

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.

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”.)

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?

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

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 the intuitive explanation of the Second Fundamental Theorem. I will ponder.

Page 54: It’s a little harder…

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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