-a tribute to Mr. Alvin Mayes who taught Calculus to me 50 years ago, and an unnamed graduate student who taught Calculus at the University of Michigan a year later-

Some of this will not be understandable by those that do not understand calculus. None of it will be understood by those who do not try.

Calculus is a language and way of thinking, a way to describe properties of the real world and to learn about them. It was first discovered by Newton and not published and then discovered by Leibniz. Although it is used for other things it provides a language for describing the properties of curves, an advanced vocabulary for some properties of solids, and ways to understand the cumulative action of forces on the movements.of objects. It has its little fingers in every aspect of our life.

I cannot say that I use calculus as much as I could or even should, but it does get used when I press the accelerator of my car. I think about it as I turn the water down as I am filling a container so that it won’t overflow. Acceleration, and deceleration are topics covered in calculus classes. When you leave a stop sign do you depress the gas pedal at a constant rate? I don’t. I press my foot down rather quickly at first because I want people to know quickly that I am moving and then slow the depression of my foot with the rate moving towards a steady state when I am at speed. I am sure that how I do this is not optimal for gas savings. How I do this is a compromise between safety, speed of travel and gas. I also try not to tax my brain too much as I drive. My understanding of this is informed by my understanding of the curves that describe the motion and its affects.

I have taught a lot of pot making over the years. It might not be the most effective way to teach about form, but for some people the discussion of form is best done classically by breaking it up into pieces that can be described. There is more than one way to cut a cake, and really you do not always even need a knife. But one of the ways I do this is to talk about how the radii that are tangent to the pots curve at different places change in length. I should be clear for the mathematically inclined that I do not discuss slope as described by my Geometry teacher Mr. Waldman as “der rise über der run” because it is not as useful as talking about a more linear seeming property, the rate of change of the angle of the slope. (By this definition a vertical line has a slope of infinity, a horizontal line 0 and a 45˚ angle one. The difference between horizontal and 45˚ with this concept is 1 and from thereto vertical is ∞ -1. This is not intuitive.) I talk about change of angle. With this a tight curve in any orientation has the same description. The radius of a curve at any point in the curve is a basic concept that I use in most other concepts about curves.

Studying calculus you look at a lot of curves, think of them in terms of rates of change, or acceleration, deceleration. When you push down the gas pedal at a constant speed your car accelerates, but there are competing properties at work. It is easiest to talk about in a manual transmission once the car is moving. At the low end, when the pedal is only depressed a little bit your car accelerates but as the pedal moves down further, if it does it fast enough you car starts to accelerate faster and faster. At some point near the “pedal to the metal” point, fully depressed, your car may still accelerate, but does so slower and slower. I have slightly redefined some concepts in calculus so that I can use them in life without having to work too hard.

If you are trying to get your car to say 60 kilometers per hour and do so quickly, you may push the pedal down rather quickly and find that you have to ease off when the car reaches sixty. At some point driving like this your foot stops going down and starts easing off. This is an inflection in the depression of your foot. Knowing where you should do this makes your driving smoother and probably reduces gas consumption. I suspect that not having an inflection at all probably saves more. Knowing where to depress your foot to to achieve a steady state and easing up on how fast your depress the pedal as you approach it also likely improves mileage. It also keeps your passengers happy.

An adult I used to drive with complained that the dog only threw up in the car when they were driving. This is because they used the accelerator similar to an on and off switch with no idea that a steady state might be better. That they used to teach calculus did not help them understand its application in driving. So it goes.

As a curve gets tighter its angular rate of change accelerates, it changes more and more quickly. As it gets broader this decelerates, changes slower and slower. When a curve goes from being concave to convex the mathematical description is that its second derivative goes from being a positive number to a negative one. The mathematical concept may not of much use to the casual user of calculus concepts. But the point of change between the concave and convex is a critical point in the description of some “complex curves” (my phrase, meaning a piece of a curve with a discontinuity in the first or second derivative or a curve that goes from concave to convex ) . This point, again where the curve goes from concave to convex is called an “inflection point”. Knowing precisely where this is in change from the belly to the neck in some jars or jugs makes recreating the object much easier.

A “first order discontinuity” (an abrupt change in the first derivative) is a place where two curves join producing a sharp angle. A “second order discontinuity” (second derivative) is a place where the radius of a curve abruptly changes. So if you take a 3 inch diameter arch and attach it directly to a 5 inch diameter arc so that there is no corner , this is a second order discontinuity. If one is concave facing up and the other is convex facing up,,,, going from concave to convex,,,,, then this is also an inflection point. A first order and inflection together is really quite unusual but does happen at the intersection of bellies of bottles to the shoulder. it can have a very different look and feel than the second order kind, you have to keep this straight. While I think about third order changes, I do not think that I can perceive them in products. I let them be as just abstract thoughts.

In throwing on the wheel the axis of rotation is always vertical. But a dear departed teacher Victor Babu, used to talk about taking a curve and slightly tipping this axis. He did not use the phrase axis of rotation. His way of describing this was in some ways better. He tipped the curve to be rotated. Not having a good phrase to hold the concept made this sometimes difficult to describe to some people.

In high school I did very little homework. This is in part because I hated writing anything. ADHD likely played into it. The “write it over stuff but make it neater earlier in my education made homework as a topic very unpleasant.” The only time I can remember completing homework regularly was when studying with a friend. I did learn. In 11th grade I was in the semifinals for the Michigan Math Prize competition. I think that this changed my standing in the eyes of Math teachers, but they already knew I was smart. I had not studied enough to have a chance at the finals. My high school owned this competition back then. It was a great place to go to school.

Because of not always getting all the concepts and not doing homework, not practicing, and errors in basic processing of numbers I got mostly B’s and C’s in math. I did get a D in 10th grade (might have been 9th grade) and my mother made me take a course over the summer. To the best of my knowledge I was the only person ever to pass Mr. Waldman’s Geometry class without doing my homework. I read the textbook over the summer before it. At the end of the first few days I took the first test and got 100 percent, then took another quiz . A few days later I took the final. Despite his policy of failing the course if you did not do homework, I got an ‘A’. Its fortunate, because if required to do homework I would have failed. Unaccommodated dysgraphia can be a disaster.

After my first class in Calculus with Mr. Mayes, a class I believe I got a ‘B’ in, I got a ‘3’ on an advanced placement test. This was not a high enough grade to earn credit for Calculus at The University of Michigan Engineering School. It did show that I knew much of the content. I had not memorized formula and rules for doing some problems. The class despite being difficult. I can remember the good nature of Mr. Mayes. He had an incredible smile an patience. But the ‘3’ meant that I had to take calculus in college. The TA was a joy. Starting at the beginning again it all quickly came into focus. I did not do the homework, but easily aced the class.