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?"