The post Cartoon Calculus: Back matter appeared first on Stand-Up Economist.

]]>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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]]>The post Cartoon Calculus: Ch 15 (Conclusion) appeared first on Stand-Up Economist.

]]>**Page 187: That was exhausting…**

**Pages 188-189: It’s tempting to think that calculus is the pinnacle… // That’s because mathematics is about finding patterns…**

**Pages 190-191: The calculations of pi in Chapter 3… // If you think those examples are trippy…**

*MB: p. 190 ? Are the right hand pictures supposed to be the limiting figures??????? How can this be? They are just figures along the way to the limits, aren’t they?*

**Pages 192-193: For another example of limits, consider prime numbers. // It turns out that these precise answers…**

*MB: p. 193 If you are going to use the symbol ln, then define it. On graph—label axes (maybe use ’n’, not ‘x’). Symbol x-> infinity should be down below. I think this example is too abstruse.*

**Pages 194-195: And of course we use limits in calculus… // …and integrals…**

**Pages 196-197: Calculus applies to rabbits because… // The mathematical result is called a differential equation…**

*MB: p. 197 This will be unintelligible to most of your readers, I would think, but if you want to use it, explain what equation is doing and define P_o*

**Pages 198-199: Another advanced topic… // Multi-variable calculus…**

*MB: p. 199 Ditto. Show a picture, define all terms (P_L, P_K) or don’t do this.*

**Page 200: In short, there’s plenty more to study…**

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

The post Cartoon Calculus: Ch 15 (Conclusion) appeared first on Stand-Up Economist.

]]>The post Cartoon Calculus: Ch 14 (Physics) appeared first on Stand-Up Economist.

]]>**Page 173: What does it mean that acceleration…**

**Pages 174-175: It’s not immediately obvious why Leibniz… // Newton’s laws of motion and gravitation…**

**Pages 176-177: Newton’s second law of motion… // That’s because we need limits to define acceleration.**

*RH: Newton’s 2nd law correctly stated says that the NET force on an object equals its mass x its acceleration. This is an important distinction from just saying F = m x a. Here’s why: A book sitting still on a table is not accelerating (a = 0) but it has two forces on it, neither one of them being zero. So, if we say F = m x a (which is NOT true) and (a = 0) this somehow implies there’s no forces acting on the object. But there are! So adding the word “net” is critical here (and takes nothing away from your story). *YB: Thanks for this suggestion; we will ponder.

*RH: You make this claim: “…AND ACCELERATION IS HARD TO UNDERSTAND WITHOUT CALCULUS.” I actually disagree with this but I think it IS fair to say that “acceleration is easier to understand WITH calculus.”* YB: I think we may just have to agree to disagree, but note (relative to the next paragraphs) that when the book talks about things like velocity and acceleration it’s referencing something instantaneous, not something averaged over time.

*RH: You state “…SO WILL OUR VELOCITY INCREASE BY 9.8 METERS PER SECOND DURING THE NEXT SECOND?” and then answer “NOT QUITE.” But that IS what the 9.8 m/s2 means! Given that the acceleration due to gravity near the surface of the earth is 9.8 m/s2, any object in free fall will increase its velocity (in the negative, downward direction, mind you) by 9.8 m/s for every second of flight, regardless of its initial velocity, be it zero, upwards, or downwards, or even any combination of those. All that the limiting process does here is zero in on the instantaneous acceleration. But in the case of free fall the acceleration is constant over time so the limit process doesn’t nearly have the same impact here as it does for speed since the instantaneous acceleration is always equal to an object’s average acceleration.* YB: Good point, we can try to clarify this (as we do in Ch 2) by saying, e.g., that your velocity might not increase by 9.8 meters per second during the next second if you open your parachute, or if you hit the ground, etc. This is what we did in Ch 2, and hopefully we have the space to do it here as well.

*RH: You also bring this issue up in CH2 Speed. When I teach physics I bring up the idea that there are different ways to measure speed: The AVERAGE SPEED of something during a given time interval is the slope of its position-time graph between two points on the graph which bookend the time interval. But if we wish to know something’s speed RIGHT NOW (a specific moment in tie) we need to find its INSTANTANEOUS SPEED. And that is where the limit ∆t 0 comes into play. [I always thought this would be a good joke in a book like yours, particularly so if you showed it with the associated position-time graph: Police Officer to a driver he just pulled over: “Your instantaneous speed just now was way above the speed limit.” DRIVER: “But officer, my AVERAGE speed was UNDER the speed limit.” Both of them can be right!] *YB: Okay, but we’re focusing just on the instantaneous speed… so the joke will have to wait for a different book.

*MB: First off, it seems as if you are referring to forces other than gravity when you say the measured acceleration after one second won’t be = g. That turns out not to be your point, but I am not sure what your point is. Yes, acceleration is defined as a derivative, i. e., a limit, but if it is constant over a second, you will indeed see delta v = 9.8 m/s. Is it important at this level to say anything else? Couldn’t you just draw a graph of x[t] and fit it by little segments that get smaller and smaller?* YB: This is similar to RH’s comments; clearly we need to address this point.

*RH: Page 177: Again, you don’t NEED limits (as stated at the top of P5) to understand acceleration but they certainly can help.* YB: Discussed above.

**Pages 178-179: In the language of mathematics… // Luckily, the math itself is pretty easy.**

*RH: I like the sequence of equations you show for the Flying Apple Example. One thought here: What if you used a small font to label what each number means? For example, for the f(t) equation the number 2 represents the initial position (in meters), the +19.6 is the initial speed (in m/s) and the – 4.9 is half the acceleration (in m/s2). These same meanings carry through the rest of the equations, too.* YB: Good ideas. I kind of doubt that we have the space, but we can try.

*MB: p 178 Here again, just going from variable t to variable x can throw some readers. Either explain or use same variable.*

*MB: p. 179 Here the 19.6 and the 2 are chosen at random—explain!*

**Pages 180-181: Of course, the pull of earth’s gravity isn’t constant. // Figuring out how to get an apple to an astronaut…**

*RH: This line: “NEWTON’S LAWS TELL US THAT IT CHANGES WITH ALTITUDE.” When people refer to “Newton’s Laws” they mean his three laws of motion, not usually his Universal Law of Gravitation, to which you refer here. Yes, it is “a law” that Newton authored but I think you should restate it is a follows “NEWTON’S LAW OF GRAVITATION TELL US THAT IT CHANGES WITH ALTITUDE.”* YB: Good point. We try to finesse this on page 3 by referencing “Newton’s laws of motion and gravitation”, and the question is whether the extra verbiage provides extra clarity or just gets in the way. We will ponder.

*RH: I think you should add the units (meters per second per second or m/s2) to the vertical axis here. It looks wrong (to me) to have units on one axis but not the other.* YB: Excellent point. We will definitely fix this. (We have a bunch of work to do on the graphs in the book more generally.)

*RH: Near the top of the page you say you are trying “to get it (the apple) up into orbit.” Our scenario: There is an astronaut 2.5 x 107 meters away from the center of the earth and you are trying to “throw” an apple up to him. This is a perfectly good problem and you guys solved it correctly provided….wait for it…the astronaut is NOT in orbit. What you have done is solved for the speed required to launch an apple up to an astronaut who sits at a distance of 2.5 x 107 meters away from the center of the earth. The key word here is “sits.” If the astronaut were “in orbit” AT THAT ALTITUDE he would be moving in a circle around the earth at a velocity of 4001 m/s (That “sideways” speed is what keeps the astronaut up there, in orbit, and prevents him from falling back to earth. Our “sitting” astronaut will immediately begin falling toward earth just like anything else dropped from rest at that altitude.) Which means that to put an apple” into orbit” you’d need to give it more energy than you calculated (about 17% more) to give it the same speed of 4001 m/s at that altitude that the astronaut had. Put another way, you are calculating the launch speed required to send an apple upward so that when it reaches a height of 2.5 x 107 meters it has no speed anymore (it begins to “nose over.”) So, what’s the easy fix here? Just DON’T use the phrase “get it up into orbit” otherwise you’ll have more explaining to do than you want. (Maybe just say “…to get it up to the astronaut.”) I do really like this example, however! *YB: Good point, we will work on fixing this language. Too bad, because it’s an evocative phrase, but you’re right that technically speaking it’s not correct.

*MB: You picked a hard problem. What about rotation of the Earth and motion of the astronaut???? The apple won’t get to the astronaut in your problem. And….use of the term ‘work’ here will be confusing. (It always is, even in more straightforward applications; the relationship of ‘work’ to forces is not intuitive to beginning students.) You just mean—how much energy must we give the apple? Use the symbol v[0] to indicate initial velocity.* YB: This is similar to RH’s comments, except for the suggestion of discussing energy instead of work. We will ponder.

**Pages 182-183: This is a tough physics problem… // …but the math boils down to this question.**

*MB: p. 182 Your dimensions of C are wrong. We discussed work. Put m_2, not just m, in equation. (And maybe use m_apple instead of m_2?)*

**Pages 184-185: We could calculate this integral the hard way… // It’s especially easy because…**

*MB: p. 185 Explain Joules.*

**Page 186: Calculus is obviously not sufficient to solve physics problems…**

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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]]>The post Cartoon Calculus: Ch 13 (The first fundamental theorem, revisited) appeared first on Stand-Up Economist.

]]>**Page 159: Introductory page**

**Pages 160-161: It would be a bit too intense… // Story #1 is about a galloping horse.**

**Pages 162-163: One way to calculate average velocity… // Fortunately, we can find this limit…**

**Pages 164-165: But in addition to calculating average velocity… // We just covered two different ways…**

**Pages 166-167: Story #2 starts when we take two basic facts about circles… // We can use the tools from Chapter 7…**

*MB: p. 166 Draw a circle and a tiny ring around it of width dr. Area of ring is 2 pi r dr.*

**Pages 168-169: The next part of story #2 is about rings. // But another way to calculate the area of a ring…**

*MB: p. 169 The area of the ring is not the sum of the circumferences!! *

**Pages 170-171: Of course, these two ways to calculate area… // This story about area isn’t a proof…**

*MB: p. 171 Your analogy isn’t quite right: area= integral of circumference times thickness. Distance = integral of velocity times time.*

**Page 172: And there’s a bonus…**

**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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]]>The post Cartoon Calculus: Ch 12 (Integration the easy way) appeared first on Stand-Up Economist.

]]>**Page 147: A journey of a thousand miles…**

*WM: Is there a reason in Chapter 12 you don’t generalize to the integral power rule the way you generalized to the derivative power rule? You’re 3/4 of the way there just given the explanations you have already written.*

**Pages 148-149: In Chapter 4, we noted… // As we’ll see in this chapter**

*MB: p. 148 Again, I think that instead of opposites, you want to say that you do something to f[x] (take its derivative) and you get f’[x]. Now what do you have to do to get back to f[x]?*

**Pages 150-151: As we’ll see, finding anti-derivative… // To see how super easy it is…**

*MB: p. 150 Good and bad anti-derivatives—did you make up these terms? They are another distraction; at this level you don’t have to emphasize peculiar functions.*

*MB: p. 151 I suggest just integrating to a particular value of x, say x_0; not a constant (1). Then clearly F[x_0] is another function. To find its derivative, imagine finding F[x_0+h]. Draw the picture. Fancy that! The derivative of F[x] is just f[x]!*

**Pages 152-153: More generally, the easy way… // This result is so important…**

**Pages 154-155: The best way to find a good anti-derivative… // In this case, trial and error…**

*MB: p. 155 Don’t forget the ‘dx’ in integral.*

**Pages 156-157: Unfortunately, there’s no foolproof method… // And every time you learn something new…**

*MB: p. 156 Why this function?*

*MB: p. 157 Misprint in first equation (x^(-2), not x^2).*

WM: Page 157 – I don’t get the Chekhov’s gun reference. [And then later: I see the Chekhov's gun thing reappears in Chapter 14, so you do go back to it.]

**Page 158: One final point…**

**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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]]>The post Cartoon Calculus: Ch 11 (Integration the hard way) appeared first on Stand-Up Economist.

]]>**Page 131: Part Three (introductory page)**

**Page 133: A journey of a thousand miles…**

*WM: I don’t have a lot of comments on Chapter 11. I think you did a perfectly good job, but this is a topic I think most Calc I students don’t really understand- they just live through it. You might tell readers they could skip this chapter if they like.*

**Pages 134-135: In Part One, we learned… // The hard way is to directly climb Mt. Integral.**

*MB: p. 134 Explain that ‘area’ is not the number of square centimeters on the drawing, but area measured in units of x and y. Label the axes. Show that your curve is y=x^2.
Then—the dx is NOT just added on for emphasis!!!!!*

**Pages 136-137: The way to directly climb Mt. Integral… // The trick is to add more and more rectangles…**

*MB: p. 137 Give each rectangle a width Delta x and a height y[x] and remind readers that area of each rectangle is y Delta x. (Not all will remember this.)
*

**Pages 138-139: Of course, there are many different paths up a mountain… // In the same way, there are…**

*RW: The disaster can be avoided by using Lebesgue integration and measure theory, so it might be funny if one of them was French. Also, since Archimedes actually computed the area under x^2 and x^3, you should probably pull him in there somewhere.*

**Pages 140-141: As an example, let’s return… // This example turns out to be lucky…**

*MB: p. 140 Unless you explain (as in previous sentence) how you are calculating these areas, your formulae will be baffling.*

*MB: p. 141-146 This long discussion is a distraction and involves drawing some conclusions out of an unknown hat. These pages could be omitted and you can just make the point that adding up all those little rectangles is tedious and never ending.*

**Pages 142-143: That example may not have seemed too hard… // That may seem impossible…**

**Pages 144-145: So let’s return to the same mountain… // Once again we’re lucky…**

**Page 146: That’s it!**

**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

The post Cartoon Calculus: Ch 11 (Integration the hard way) appeared first on Stand-Up Economist.

]]>The post Cartoon Calculus: Ch 10 (Economics) appeared first on Stand-Up Economist.

]]>**Page 119: Introductory page**

**Pages 120-121: Economics is about… // …and that makes calculus extremely handy…**

**Pages 122-123: What makes optimization problems in economics special… // The way economists balance them…**

*MB: p.122 Define Q.*

*WM: Page 122 – I would use the gag “Isn’t that special?”, but younger readers won’t get the joke, and Dana Carvey would sue you.*

*MB: p. 123 Maybe pick different numbers so the 2 Q^2 doesn’t come in on both sides?*

**Pages 124-125: A mathematician would set up this problem… // An economist would head for the same destination…**

**Pages 126-127: The way for you to find economics enlightenment… // Now let’s apply the same logic…**

*MB: p. 127 Explicitly define marginal cost and marginal revenue.*

*WM: Good explanation on page 127.*

**Pages 128-129: Similar examples of marginal analysis… // Of course, we shouldn’t forget…**

**Page 130: When you encounter more economics…**

**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

The post Cartoon Calculus: Ch 10 (Economics) appeared first on Stand-Up Economist.

]]>The post Cartoon Calculus: Ch 9 (Optimization) appeared first on Stand-Up Economist.

]]>**Page 107: Help**

**Pages 108-109: Finding extreme values is important in economics… // In this chapter we’re going to use calculus…**

**Pages 110-111: Our first challenge here is to translate words… // For our problem, it’s easiest to focus…**

**Pages 112-113: It often helps to sketch a graph… // and in this case the graph’s symmetry…**

**Pages 114-115: In the last chapter we learned… // So we take our function…**

*WM: Page 115- maybe have someone thinking “L is my variable this time instead of x, but the formulas work in exactly the same way.”*

**Pages 116-117: Thanks to calculus… // Then all we have to do…**

**Page 118: The world is full of optimization problems like this…**

**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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]]>The post Cartoon Calculus: Ch 8 (Extreme values) appeared first on Stand-Up Economist.

]]>**Page 95: Introductory page**

**Pages 96-97: Recall from page 24… // And because the apple stops…**

*WM: Page 97 – I would write out the f(2) calculation as “2 + 19.6*2+ 4.9*2^2=” just to make clear you are going back to the function def. from the previous page.*

**Pages 98-99: Imagine that you’re taking a hike… // By analogy, imagine that you have a function…**

**Pages 100-101: On your hike… // Similarly, the extreme values of a function…**

*MB: p.100 Maxima and minima are pretty intuitive—maybe this discussion can be shortened to simply show, in a picture, that the location of these corresponds to places where derivative =0. (And hint at second derivative?)*

**Pages 102-103: But on many hikes… // Similar, many functions have extreme values…**

*MB: p.102 etc. Optimization. Emphasize in words that two tendencies (growing height or growing width) fight each other and at optimum they balance.*

**Pages 104-105: Okay, let’s summarize. // Calculus allows us to eliminate…**

**Page 106: In fact, calculus makes this process so simple…**

**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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]]>The post Cartoon Calculus: Ch 7 (The calculus toolkit) appeared first on Stand-Up Economist.

]]>**Page 83: (Introductory page)**

**Pages 84-85: Some fruits are easy to eat… // What you need are the right tools.**

*MB: explain why this particular function? And if the variable should be time, as I suspect, then make it t, not x, which is distance in this text.*

**Pages 86-87: Let’s start with the sum rule… // The sum rule also makes sense…**

**Pages 88-89: Next, let’s look at the constant multiple rule… // The constant multiple rule also makes sense…**

*MB: p88: State that you will use the letter ‘c’ for a constant, or number, that does not change.*

**Pages 90-91: Somewhat more complicated is the product rule… // As x changes…**

*WM: Page 91 – very nice.*

**Pages 92-93: There are lots of other rules… // Luckily, we’ve already completed step 1…**

*MB: p92: I am not sure that in this text you need such a long derivation.*

*WM: Page 93 – maybe move the “it may help” statement down in the picture to that third step where you actually use it.*

**Page 94: Thanks to the calculus toolkit…**

*MB: p. 94 Misprint in top equation. Perhaps tell us what you are doing in terms of distance, velocity, etc., before you do it. The whole discussion, p. 94-97, should be accompanied by a picture of the trajectory and an explanation of what the various functions and derivatives are on this picture.*

**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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