So What Is Calculus Already?
Calculus is basically just very advanced algebra and geometry. In one sense, it’s not even a new subject — it takes the ordinary rules of algebra and geometry and tweaks them so that they can be used on more complicated problems. (The rub, of course, is that darn other sense in which it is a new and more difficult subject.)
Look at Figure 1-1. On the left is a man pushing a crate up a straight incline. On the right, the man is pushing the same crate up a curving incline. The problem, in both cases, is to determine the amount of energy required to push the crate to the top. You can do the problem on the left with regular math. For the one on the right, you need calculus (if you don’t know the physics shortcuts).
For the straight incline, the man pushes with an unchanging force, and the crate goes up the incline at an unchanging speed. With some simple physics formulas and regular math (including algebra and trig), you can compute how many calories of energy are required to push the crate up the incline. Note that the amount of energy expended each second remains the same.
For the curving incline, on the other hand, things are constantly changing. The steepness of the incline is changing — and it’s not like it’s one steepness for the first 3 feet and then a different steepness for the next 3 — it’s constantly changing. And the man pushes with a constantly changing force — the steeper the incline, the harder the push. As a result, the amount of energy expended is also changing, not just every second or thousandth of a second, but constantly, from one moment to the next. That’s what makes it a calculus problem. It should come as no surprise to you, then, that calculus is called “the mathematics of change.” Calculus takes the regular rules of math and applies them to fluid, evolving problems.
For the curving incline problem, the physics formulas remain the same, and the algebra and trig you use stay the same. The difference is that — in contrast to the straight incline problem, which you can sort of do in a single shot — you’ve got to break up the curving incline problem into small chunks and do each chunk separately. Figure 1-2 shows a small portion of the curving incline blown up to several times its size.
When you zoom in far enough, the small length of the curving incline becomes practically straight. Then you can solve that small chunk just like the straight incline problem. Each small chunk can be solved the same way, and then you just add up all the chunks.
That’s calculus in a nutshell. It takes a problem that can’t be done with regular math because things are constantly changing — the changing quantities show up on a graph as curves — it zooms in on the curve till it becomes straight, and then it finishes off the problem with regular math.
What makes calculus such a fantastic achievement is that it does what seems impossible: it zooms in infinitely. As a matter of fact, everything in calculus involves infinity in one way or another, because if something is constantly changing, it’s changing infinitely often from each infinitesimal moment to the next.
Real-World Examples of Calculus
So, with regular math you can do the straight incline problem; with calculus you can do the curving incline problem. With regular math you can determine the length of a buried cable that runs diagonally from one corner of a park to the other (remember the Pythagorean Theorem?). With calculus you can determine the length of a cable hung between two towers that has the shape of a catenary (which is different, by the way, from a simple circular arc or a parabola). Knowing the exact length is of obvious importance to a power company planning hundreds of miles of new electric cable.
You can calculate the area of the flat roof of a home with regular math. With calculus you can compute the area of a complicated, nonspherical shape like the dome of the Minneapolis Metrodome. Architects need to know the dome’s area to determine the cost of materials and to figure the weight of the dome (with and without snow on it). The weight, of course, is needed for planning the strength of the supporting structure.
With regular math and simple physics, you can calculate how much a quarterback must lead a pass receiver if the receiver runs in a straight line and at a constant speed. But when NASA, in 1975, calculated the necessary “lead” for aiming the Viking I at Mars, it needed calculus because both the Earth and Mars travel on elliptical orbits, and the speeds of both are constantly changing — not to mention the fact that on its way to Mars, the spacecraft was affected by the different and constantly changing gravitational pulls of the Earth, moon, Mars, the sun. See Figure 1-3.
