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

Table of Contents

General comments here!

*RB: On first impression, I like the mountain motif! It brings to mind the approach of https://www.youtube.com/watch?v=OmJ-4B-mS-Y (Map of Mathematics), and adds a topology.* YB: This is a neat video!

*RW: This is fun! I’m a little confused about the audience age. Sometimes it seems its for HS and sometimes for college. Generally i don’t have a lot of comments – I used to teach this stuff and I found it both entertaining and enlightening.* YB: Thanks!

*DLJ: There are two important things about limits, it’s important to keep them separate, and most textbooks don’t.*

~~Important Thing Number One is that Bishop Berkeley objected to the whole calculus proposition, because infinitesimals can’t exist, and a whole lot of other hooey. The debate went on for decades, and “at the limit” was the verbal formula that eventually made all the Berkeley types shut up. This is a historical curiosity, or horror-story, all PhD’s know about it, and when they get around to writing calculus texts, they natter on about limits for ever and ever because they have this fixation that they’re historically important. Ugh.~~

~~Important Thing Number Two is that in this world we can never know anything with certainty. Certainty is the business of belief, and hence of religion. Science is the best we can do with the limited knowledge we have and our feeble powers of reason, so it’s naturally never certain. Therefore we’re always doing the best we can with whatever it is we’ve got. ~~

~~(The brilliant Herb Simon, winner of one of the early Nobel Prizes in Economics [oops, you knew that...] coined the term “satisficing” in the hope that he could make the world do better policy by not forever and always striving for perfect policy. Great idea. Complete failure!)~~

~~It’s OK for the kids to know about all that Bishop Berkeley stuff in the abstract, but they should not have their time wasted on a whole lot of uphill-downhill reworking the debates of 300 years ago. What they need to know is, yes, there’s a big philosophical point there, but here, in this subject, we’re going with the best approximation we can get.~~ ~~ YB: I think we touch on this the right amount, with a nod to Berkeley (“ghosts of departed quantities”) but not too much detail. Hope you agree!~~

~~ ~~*WM: I’ve read the pdf file, and I mainly have things I like; I only have a few minor recommendations. 1) I enjoy seeing your writing process, that is, all of the notes to yourself about what you might want added to the artwork. It’s interesting to get a behind the scenes look.* ~~ YB: Good!~~

~~ ~~*BG: Thanks so much – this looks like fun. Here is my amateur feedback. Thanks! *

* RB: First, I want to apologise. I often have a writing style that wanders between bizarre and incomprehensible, and I appreciate your kind words about my prior contributions.*

That said, I go on:

The draft led me to consider what I’d like to dub the Pizza Theorem of Derived Motivation. If at the outset of learning about Pi we know it means something to us, we may as readers be more interested.

Why do we care how many decimal points of Pi we have?

There are actual reasons. Take the example of pizza.

The crust of pizza is made of malleable, rising bread. Dough doesn’t care much about size. If you know pi to one decimal point, somewhere near 3, you can calculate how much pizza dough you will need to make a pizza.

Next, look at the sausage slices you put on your pizza: sausage stuffing doesn’t rise but casing is elastic, so you will need only about two or more decimals of accuracy to know how much stuffing to put into the sausage casing for the pepperoni on your pizza. You do like good pizza, right?

Next, let’s consider cheese. Cheese melts and toasts on top of a pizza, but a cheese wheel is a certain diameter, and you need to know how much pizza you can make from each wheel to know how many wheels to order, at the very least your cheese accuracy needs three digits of accuracy.

Tomato sauce? Comes in cans, unless you make it fresh yourself, in which case it is in pots of tomato sauce, and since the sauce is thinner than the cheese, you need four digits of accuracy in measuring your can or pot to know the right amount of tomato sauce.

Not a lot of recipes go in for more than four digits accuracy, unless you’re talking molecular cuisine; but if you are being treated to molecular pizza, five or six digits of accuracy makes all the difference.

And what about the metallurgy of the pizza oven – six or seven or more digits of accuracy – and the timing of the alternating current in its power supply – eight or nine digits of accuracy of Pi.

And the Internet connection you use to order your pizza has an information transfer rate limited by Shannon’s Law, which uses the slices of a wave form – based on Pi – to determine how much information your signal carrier can carry to and from the pizzeria, so ten, twenty, thirty or more digits of accuracy for Pi becomes important in your day-to-day life, just to order pizza.

This is why we care in a practical way how many decimal points of Pi we know, and why we care how to get a more accurate value for the ratio of circumference to radius.

One of the most valuable things I find in your books is that I know what motivates me to learn what you’re explaining; you’re really good at keeping that motivation clear in the reader’s mind in large ways and in small.

From the example of speed in a car: Neil deGrasse Tyson has a powerful discussion of “What speed is safe?” in a car, where he compares the speed of a person sprinting headlong into a wall and how much that would hurt to the speed of a car in a collision. Instantaneous change in displacement really matters, because it can hurt. Engineers do calculus to know how fast a car can turn a corner at, from how fast a car can come to a complete stop within reaction time, how much power to build into an engine to get a car from zero to the safe driving speed for a smooth commute. If we didn’t have Calculus, then Tesla would have had to start from where Henry Ford was 100 years ago to come up with these answers by trial and error, and it would take 100 years to design a new car. The ability to take what we know and process it through Calculus into useful, practical answers is a powerful motivator.

In Economics, from Adam Smith’s ideas about landlords receiving rent to limit scarcity of fruits of their land and avoid the Tragedy of the Commons by making it harder and harder to buy something as it becomes scarcer and scarcer, to Ricardo’s arguments about benefits of trade as comparative advantage fills out a possibility frontier at the limit of productivity, to the Laffer Curve expressing the shape of the limit of government revenues are all powerful practical applications of Calculus for letting us know the shape of possibilities in our world, and how to tell a good proposal from a bad one without going through the risk and expense of trying every idea to find out if it will be good or bad. In that way, we can improve management of our economies with agility and assurance, while avoiding the pitfalls of truly terrible ideas. In theory. Winking smile

*I’ll try to limit my observations going forward a little more.*

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