Cartoon Calculus: Ch 2 (Speed)

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 Newton and Leibniz hanging around and bickering throughout Chapter 2. YB: Thanks!

MT: we think that most students who take calc already have a pretty thorough understanding of speed, but nevertheless the chapter covers material that may be new to some students. For example: tangent lines, instantaneous speed, and differential calculus… Overall, Chapter 2 strikes the delicate balance between presenting new material while also still being engaging. YB: Good!

MT: For Chapter 3 (area): We like that the chapter starts with material that was covered in the previous chapter and builds on to what students already know or have just learned. We also really like how on page 31 the central idea of the chapter is explicitly stated and presented; this gives students (including ourselves) a clear understanding of what is about to be covered in this chapter–perhaps a “the big idea here is” statement could be included in chapter 2, too? YB: Good idea, but I don’t really know how to do that, sorry.

MT: We appreciate that “tangent lines” is defined in the glossary, but think “instantaneous speed” and “differential calculus” should also be defined. YB: We will do this! (Comment moved to glossary.)

Page 15: I have this dream…

RS: the opening panel with the psychiatrist couch is cute, but it’s an outdated stereotype and a dismissive of therapy. Many of students who are going to feel the need to buy a cartoon guide to calculus may be struggling with their own mental health. If you want to make a joke at someone’s expense, I’d recommend a hippie saying “what does it all mean? YB: We didn’t intend to dismiss anything or joke at anybody’s expense, so we’ll take another look.

Page 16-17: Nothing seems simpler than speed… // …but what exactly does speed measure?

RS: I think at the bottom of page 17 (the first panel with the cop) maybe you could insert something about the casual definition of speed. “60 miles an hour” means, to most people, that “if nothing else happens, I’m going to cover sixty miles in the next hour.” But really, does nothing else happen? The speedometer can’t predict the future. You’re going to slow down to look at the scenery, you’re going to get to a big downhill. I think it might be a bit cleaner to get to the notion of an instanteous rate of change by saying that the speedometer is a prediction, and therefore an approximation. This might help lead in to the mean value theorem more easily later. YB: Yeah, this is tricky stuff. I’m not sure how to improve what’s there, though.

Pages 18-19: The most obvious interpretation of 60 miles an hour… // We can try to zero in on the answer…

MB: On p. 18 and following you are using speed to illustrate the instantaneous nature of the derivative. You should add, though, that if speed is constant then indeed x = vt, because on p. 22 and thereafter you again use speed as the (constant) slope of x vs t. YB: No space I’m afraid.

Pages 20-21: The truth is that saying what speed is not… // Trying to figure it all out…

BG: do you want to give Isaac and Gottfried name tags? YB: I think their height and hair color distinguish them, but perhaps we should try to identify them elsewhere. I will suggest to Grady.

RS: On page 21, the mathematicians arguing about infitesimals would benefit from some handwaiving that some people said “it doesn’t really matter.” In fact Leibniz notation is an interesting irony of mathematics, because Leibniz was obsessive about symbols and yet his notion of dx/dy doesn’t really mean division. He didn’t seem to know that and the fact that we use it today is sort of an artifact of history. YB: I’d disagree that it doesn’t really matter. I guess I’m enough of a theorist to think that rigor does matter! RS: I was unclear, sorry. It *does* matter. At the time, though, and for hundreds of years, there wasn’t much rigor around calculus. When Leibniz wrote “dy/dx” he was clearly thinking about division. And it’s useful for students to be able to manipulate the symbols in a way that feels like algebra. I’ll think about it some more.

MT: At the top of page 21, we suggest that the four characters make the comic busy, and the fact that the characters are bringing up unknown terms adds confusion to the comic. Therefore, we recommend that the fourth character from the left (who says “ghosts of departed quantities?”) could perhaps be removed or toned down to make the comic more streamlined. YB: Can’t do this because “ghosts of departed quantities” is a historical reference to one of the early critics of calculus.

Pages 22-23: Consider a car moving at a constant speed. // In those examples, the slope doesn’t change…

BG: nice looking cars, I like the snail minivan. YB: Thanks!

RS: Your first graph appears on Page 22. The angle of the car is quite confusing. Some readers are going to try to overcomplicate things with the sense of how hard it is to drive uphill, and the notion that you’d go slower uphill. I would suggest starting with a one dimensional graph of the car at point A and then the car at point B, with a clock on each.

Then when you draw graphs, do it on an overhead view of the cars, not a side view. And just put ONE car on the first graph, and one car on the second graph. Then you can combine them in the future.

In general, slope as a concept is going to make people think about what it’s like to drive up and down slopes, and that’s going to be confusing.

The first graph where the line has some curves is an opportunity to explain what it’s like to go backwards. You could tell a story where the road trip starts, then there is construction, then there is an overlook so it’s flat as a board. Then they head off and realize they left little Jimmy behind at the overlook and have to backtrack, etc. YB: Good comments, I will discuss with Grady. The challenge is how to connect the slope with something graphical without giving the misimpression (as you point out) that a steeper slope means going steeper uphill.

WM: I would still recommend on page 22, that you have a character somewhere thinking “Slope? Oh, yeah, that’s just rise/run” with an appropriate drawing. (It’s my least favorite definition of slope, but it seems to be the one being taught incessantly, so it’s the one I recommend you use.) YB: I will suggest this to Grady, but it might end up in the glossary.

Pages 24-25: An apple thrown straight up in the air… // …and we can see those speed changes…

WM: Maybe define what a tangent line is before its appearance in the graph in Chapter 2? My students have typically never heard the word “tangent” till Calculus. YB: Good suggestion, we will try to either say more here or refer readers to the glossary.

RS: When you get to the parabola for the apple on page ~24, again I’m concerned that people will be adding a second spatial dimension in their mind. I would add little stopwatches or your mini Leibniz/Newton looking out at each point. And I would also consider a 1-d graph first. YB: Not sure we have space, but I will ponder. Update: I think we try to address this by having the graph on p25 parallel the images on p24, with the apple getting thrown straight up in the air. I think that’s the best we can do.

RS: On page 26, your selection of points looks more like 7 meters to me. I’d also have your little labcoat people holding up curly braces { } or some instrument with faint dotted lines. You can have one on a ladder if you like.

You also might consider picking an example that has both the numerator and the denominator as more than 1 so that the reader has to do a bit of arithmetic. Otherwise they might think you’re just copying the “5” from “The position changes by about 5 meters…” Same goes for the subsequent few pages. YB: Yes we need to fix these graphs, good point.

MT: We love the tangent line explanation (pages 25-28), but suggest that a few application problems could be included at the end of Chp. 2 to reinforce what was just learned. This would allow students who are confident in their calc abilities to simply skip to the end of Chp. 2 and just complete the application problems to ensure that they know the material well. Just a suggestion–let us know what you both think. YB: Sorry, but we don’t really have room in the books for problems.

Pages 26-27: We can try to zero in on the answer… // The more we zero in…

BG: all 3 drawings – you could drop a line (maybe in light grey to keep it minimal) from the secant points to the x-axis to visually show the length of time being measured is decreasing. Another way to do this w/o too much added visual info is to put black tick marks on the x-axis under the secant points, with no line. YB: Good suggestion, we will try this!

RS: Finally, on page 27 you say “we will never be able to calculate the slope of the tangent with absolute precision.” That’s not true. Eventually we’ll have a formula that describes the motion (just a 2nd degree polynomial) and we will be able to calculate the slope of the tangent line anywhere by taking the derivative. I think you need to say “we will never get there by measuring more and more closely. We need a better tool…” YB: Here I think I disagree. The point we’re making is that approximations will never get you to the actual value. RS: Agreed, but that’s not what the text says. On page 27 you I read “we will never be able to calculate the slope of the tangent with absolute precision.” I want you to add using approximations to the end of that sentence. And maybe a little asterisk and a note about how you can exactly calculate the slope of the tangent at any point—just not with approximations!. YB: The text actually says: “The more we zero in, the more precise our estimate will be… but we’ll never be able to calculate the slope of the tangent with absolute precision.” I don’t think we need to add anything at the end of the sentence because there’s something about this at the beginning of the sentence! RS: I’m trying to think about it from the perspective of someone who doesn’t know anything about calculus. If you said to someone “the more we do X, the more we’ll get closer to Y, but we will never be able to get Y” it makes it seem like Y is impossible. But Y is totally possible. It just can’t be found through technique X. See why I brought it up? YB: Let me also emphasize the need for parsimony in these cartoon books. You write “I want you to add using approximations to the end of that sentence. And maybe a little asterisk and a note about how you can exactly calculate the slope of the tangent at any point—just not with approximations!” And that’s a totally reasonable suggestion, and we will ponder, but adding asterisks and notes (or even just two more words) is not easy in a cartoon book. Hope that makes sense at least as an explanation for why I’m giving you some pushback and not just saying “Yes, good point!” RS: Understood. I want to find a wording that is (a) concise (b) engaging and (c) accurate. I think you’ve got (a) and (b), but not (c) completely. Maybe there is a rewording that does all three?

Page 28: This similarity shouldn’t be surprising…

WM: So, the point of Chapter 2 is that we need to be able to calculate the slope of the tangent line to calculate speed, and you’ll do that (I assume) once you define the derivative. Maybe tell the reader in which chapter you’ll return to this? YB: Good idea. Maybe we can do this on p28, changing “We’ll learn more about that soon” to “We’ll learn more about that in Chapter 6″?

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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