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!

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RS: the opening panel with the psychiatrist couch is cute, but it’s an outdated stereotype and a dismissive of therapy.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. We’ll ponder this.

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.

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: We’ll definitely do something to distinguish them. I’ll suggest name tags 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.

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 the left little Jimmy behind at the overlook and have to backtrack, etc. YB: Good comments, I will discuss with Grady.

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.)

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.

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.

RS: On page 25, 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.

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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