What is Calculus? Part Two

In our first installment, we saw a bit of history, and began to think about the idea of rate of change (which can be visualized as the slope of a graph). Slopes of straight lines, central to algebra, are defined by rise over run, or the ratio of change in y over change in x. In calculus, we extend our definitions so that we can consider slopes of curvy lines. Since the slope is different for different parts of the graph, we can think of it as a new function. For each x value, there is a y value, and there is a slope. We call this new slope function the derivative of our original function. (This is the part of calculus called 'differential calculus' - differentiating means finding the derivative.)

We need an example for this to make sense. Let's go back to that ball. From part one:

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 - [its] horizontal axis will measure time ... in fractions of a second... 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 (new version):

We see from the graph that the slope is positive until we reach the highest point, and after that it's negative. In fact, finding the highest point turns into finding out when the derivative (slope function) equals zero. This is a powerful tool for finding the most or least of anything that's defined as a function.

So how do we find these slopes? If I asked you to draw a tangent to this parabola at t=0.4 seconds, you would probably know what to do, even though the only definition of tangent line you may have seen before this is the tangent to a circle. Can you draw that tangent line? (I've drawn a picture below, but I'll discuss our next step now, so the picture won't show up right away. Please turn away from your computer, and draw a sketch of this graph with a tangent line added in.)

If we want a more precise description of what's happening to the ball, we need an equation. In this case, height in feet is determined by time in seconds, by h(t) = -16t²+32t+4. Notice that height at time 0 would be 4, as given in the story above. (The -16 is determined by the force of gravity on earth, and would be different under different gravitational conditions. This part can be understood more deeply by the end of a calculus course. The 32 is determined by how hard I throw the ball upward.)

The tangent line touches the graph at just one point. Using our equation for the curve, we can easily find the y-coordinate (height) at that point, but how would we find the slope? Slope requires finding the change in y and change in x, which takes two points. But all we have is one point. We're stuck.

One strategy mathematicians use might be called wishful thinking. Since we need two points to find the slope, let's pretend we have them - let's cheat! If we use a second point on our graph very near the first, and make a line that goes through both points, our line, called a secant, is pretty close to the tangent line we're seeking. What if we try to get closer and closer this way?

You may want to experiment here. Our one point is (0.4,14.24). If we use t=0.5 for our second point, we get (0.5,16). Now we can figure the slope of our tangent secant line. Slope is the ratio of change in y over change in x. We find those changes by subtracting, so we get:

That would be 17.6 feet per second. But that secant line is just a little bit less steep than the tangent line. We could get a better estimate by trying a second point at t=0.41. But we'll never get a perfect answer. Hmm ...

Ok, this is where it starts getting a little weird. We want to imagine the second point getting infinitely close to the first. The use of infinity is what separates calculus from algebra. Newton called his infinitely small quantities 'fluxions', and another mathematician of the time (George Berkeley) complained bitterly:

“What are these Fluxions? The Velocities of evanescent Increments. And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”

It was more than a hundred years later before mathematicians were able to develop a solid logical basis for calculus!  But if we're willing to trust our intuition, like Newton did, this method of taking points "infinitely close" to one another works amazingly well. If we call the distance between the time values h, and imagine h getting smaller and smaller, we would say we're "taking the limit as h goes to 0".

The algebra gets very ugly here, and it turns out it's easier to see for a generic time t than for a specific time (like t=0.4). We're imagining that t represents one time, and t+h represents another time very close to it. The height at time t+h would be

Still ugly. And it's going to get a bit worse before it gets better. Now we find the slope:

Here's where the logic gets tricky. We can't have zero on the bottom of the fraction (dividing by zero just doesn't work), but if h doesn't equal zero, no matter how close to zero it is, we can cancel it. What we're left with includes -16h, but h is so close to zero that this term doesn't really matter.  So we get that the slope is simply -32t+32 or 32(1-t). So the slope of the tangent line at time 0.4 is 21.6. We could also say that the ball is moving upward at that moment at 21.6 feet per second.

Finding an average speed (I traveled 122 miles in 2 hours, for an average of 61 miles an hour) is straightforward. It's a ratio of distance traveled over the time it took. But finding an instantaneous speed - how fast I was going at a particular moment - that takes calculus.

I think this is incredibly beautiful, and every semester I relish helping students understand this concept. But this is the first time I've written it all out without feedback from the 'students'. It's time to ask: "Any questions?"

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.

A math circle...

I mentioned Pythagorean triples to my pre-calc students last month, and told them if enough of them were interested, I would run a math circle on this topic. Ten of them signed up, six came the first week, and three came to the second and third sessions. It's small, but it's going very well, and I may do something bigger next semester.

This is my first time doing an ongoing math circle with many sessions devoted to one topic. It's also my first time getting my own students to come to a math circle. I am really happy that they keep coming. I originally said it would be five sessions, but I can see that we could easily go for six to eight sessions on the questions raised here. (I may let them talk me into extending it.)

I love it when the way they approach a problem is different than the way I would have done it. X saw a pattern I had not seen before, and we explored her pattern at length in the second session. I haven't had time to write up the details, and have probably forgotten much of it.


Week One
What examples can we come up with?  (3-4-5, 5-12-13, ...)
6-8-10 leads us to define primitive Pythagorean triples (in which gcf(a,b,c)=1; 6-8-10 isn't primitive)
Maybe writing a list of all the perfect squares up to 400 will help us find more.
What patterns do we see?

• Odd + Even = Odd
• Middle number is a multiple of 4
• c = b+1 (after which I added 8-15-17 to our list)

Week Two
One person was new, so we reviewed our first week's work for him.

We explored the "family" of triples with c = b+1. a² + b² = (b+1)² becomes a² = (b+1)² - b²
= b² + 2b + 1 - b² = 2b-1. If  a² = 2b-1, then b = (a²+1)/2. This will be a whole number whenever a is an odd number. So we got lots more: 7-24-25, 9-40-41, ...

X noticed that in the triples

3-4-5
5-12-13
7-24-25
9-40-41

the second number is 4*1, 4*3, 4*6, 4*10, ... For the nth one, we use 4 times a number n more than the previous one. I showed them why these (1, 3, 6, 10, ...) are called triangle numbers, and asked them to add 1 to 100. They each came up with their own way of thinking about it. We came back to X's pattern and wrote:

a=2n+1
b=4*n(n+1)/2=2n(n+1)
c=b+1

Week Three
Another new person came, so we summarized for her. Then we explored triples where c = b+2.

I love seeing their creativity and persistence. At the same time, I am blown away by the holes in their understanding of algebra moves. Y was considering (4n)², and thought he might have to distribute.

We verified that we get all of the primitive Pythagorean triples with c=b+2 using:

a=4n
b=4n²-1
c=4²+1

Not sure where we'll take it in Week Four, but eager to find out. I am still struggling to lead less, become less visible, and listen more.

Playing With Geometry

The Math Monday Blog Hop, at the love2learn2day blog, has a theme each week. This week, it's Geometry. At community college, we don't teach geometry as a course - not surprisingly, I feel it's my weakest area in math. So I love playing with games, puzzles, and problems that stretch me geometrically.

A Game
My favorite geometry puzzle/game is Katamino, which makes a two-player game out of pentominoes (I blogged about it here). I love the beautiful wood pieces. (I can't tell if the newer versions are wood or plastic. I recommend trying to find the wood.) Playing with this will work your visualization skills.

A Puzzle
Online, I recently stumbled on a site which offers 40 challenge problems in compass and straightedge geometry, implemented in an online puzzle.

Some Problems
I've recently come to love the Five Triangles blog, where mostly geometric problems are posted a few times a week. Their problems often challenge me to think in slightly new directions.


What's your favorite way to play with geometry?

Playing With Math - The Book

I've been working on Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers for almost five years now. For years I've been saying "It's almost done." I have sometimes felt bad about how long it has dragged out. I'm not sure if the authors (over thirty of them!) trust that the book will really get into print. It will.

Last week I was perusing The World of Mathematics, by James Newman (a lovely four-volume mathematical encyclopedia), so I could write a description of it for my Book Picks section in Playing With Math. He mentioned in the preface that it took him fifteen years to put it together! Ok, now I don't feel so bad.

I've been sending out little updates about my progress on my facebook page for Playing With Math, and mentioned that there today. If you'd like to follow my progress, please 'like' that page.

Now I have something slightly bigger to say, which seems to fit better here on my blog. I'm working on answers to the puzzles Paul Salomon created - his Imbalance Problems. I am stuck on the last one we included. It may be that my brain is fried, and you'll all see an answer easily. But I just don't.

This puzzle was created by one of Paul's fifth grade students, Felix. Can you solve it?

Now I wish I had used these at the beginning of our inequalities unit in pre-calc. They really make you think about what > and < mean! I think I'll hand them out anyway - as a challenge.