Monday, October 28, 2013

What is Calculus? Part One

The New York Times has an online column called Numberplay, written each Monday by Gary Antonick. It usually features mathematical puzzles, but this week he was asked to explain 'differential calculus', and in his article he wrote:
I’ll lay out a few of my own thoughts but I’m especially looking forward to learning from others on this. Let’s make differential calculus our puzzle for the week.

He shares a delightful story of working in some very eccentric bakeries. Delightful though it was, I didn't feel like it would help anyone who didn't already get something about calculus. I want to offer up just a tiny bit of history, and then see if I can explain this for people who have at least a rudimentary understanding of algebra.

Isaac Newton and Gottfried Leibniz both invented calculus at about the same time. That sort of thing happens often in math. I think if neither of them had been around, someone else would have invented the methods of calculus - the problems being investigated by scientists of the time were just crying out it for it. Planetary orbits, gravity, the paths of projectiles, the shapes of lenses for microscopes and telescopes - understanding any of these requires calculus. (See the first six pages of this book.) The time was ripe.

Calculus mainly addresses two sorts of questions: How fast something is changing, and finding areas of irregular shapes. (The methods of calculus can also find volumes and the lengths of curves, among other things.)  Addressing these questions requires dealing carefully with the notion of infinity, mostly what it might mean to be infinitely close and what an infinitely thin slice might mean.  The Greeks (and probably others I don't know enough about) did some good work with these ideas. Even though they lacked the super-versatile tool of algebra, they made a good start on the area problem. Perhaps through lack of interest, they don't seem to have addressed the rate of change problem. Other cultures - and other individuals before Newton and Leibniz - made progress on both questions. What Newton and Leibniz did that got them famous was to find a simple link between the two, which is now called The Fundamental Theorem of Calculus.

The rate of change questions are addressed by what's called 'differential calculus' and the area questions are addressed by what's called 'integral calculus'. Since the question asked in Numberplay was about differential calculus, we'll start there. Even though the ancients did more with areas, rates of change are where we start in calculus courses these days.

If I made 3 loaves of bread a day (Are Gary and I making you hungry?) and saved them all, I could show how many loaves would be in my pantry with a simple graph, something like this ...

In algebra, we talk about the slopes of straight lines like this. For every one unit you go over, you need to go up three units to stay on the line, so we say the slope is 3. (You might remember "rise over run" - that's part of the definition of the concept of slope.) Well, slope and rate of change are pretty similar ideas. Rate of change just gets a bit more precise about the story that goes with the graph. In this case, we're adding 3 loaves of bread per day to the pantry.

The problem is, this notion of slope or rate of change only works with straight lines and constant rates of change, but most of the interesting things scientists want to study have changing rates of change. Like when we toss something into the air, and gravity pulls it back down. The methods of calculus helped Newton to understand how gravity affects objects, and it's a good place for us to start too.

Suppose we throw a small ball straight up, and are able to measure its position perfectly every tenth of a second. We can make a graph for that, too. Like our bread graph, the horizontal axis will measure time, but now it will be in fractions of a second instead of in days. The vertical axis will simply measure the ball's height. It starts out in my hand, about 4 feet off the ground. I think I can throw it about 20 feet high. We'll imagine together that I do. Working out what gravity does to that ball is one of the things we can do with calculus. Here's the graph. (I don't like that I showed the curve for negative time values. I wonder if I can fix that.)  I'll explain this one in part two.


  1. Sometimes it's the students who give us a great entryway...A student of mine asked me a great question. "Mr. Abdulla. I get the idea of average velocity, but what do you mean when you say something has a velocity of 2.5 m/s at exactly 2 seconds. I mean, at exactly at 2 seconds, like at that exact moment doesn't make any sense." To actually answer that question requires us to define a derivative much the way Leibniz did, if I remember correctly

  2. Ecept that before the Mayor of Amsterdam, Johann Hudde, found tangents and maxima without limits. See "The Lost Calculus (1637-1670), Tangency and Optimization without Limits", by Jeff Suzuki in the MAA Mathematics Magazine, Dec, 2005. A virtual unknown of whom it is told, "Leibniz in particular was impressed with Hudde’s work, and when Johann Bernoulli proposed the brachistochrone problem, Leibniz lamented:
    If Huygens lived and was healthy, the man would rest, except to solve your problem. Now there is no one to expect a quick solution from, except for the Marquis de l’H╦ćopital, your brother [Jacob Bernoulli], and Newton, and to this list we might add Hudde, the Mayor of Amsterdam, except that some time ago he put aside these pursuits ."
    By the way, Calculus teachers who dismiss l'Hoptial for puchasing Bernoulli's work might re-examine their position in light of such praise from Leibniz.

  3. Hi Sue, great topic of discussion. I just wondered what you tell your students? Do you tell them that integral calculus is about infinite summations or about finding areas and volumes?

  4. Or?

    I tell them it's about finding areas and volumes, and the powerful tool it uses is summation. The idea of infinitely close points from derivatives becomes the idea of infinitely thin slices.

  5. Here's the link to the fascinating article Pat references above:

    Thanks, Pat!


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