An Overview of Calculus

 What Is Calculus?

In this chapter, I answer the question “What is calculus?” in plain English, and I give you real-world examples of how calculus is used. After reading this and the following two short chapters, you will understand what calculus is all about. But here’s a twist: Why don’t you start out on the wrong foot by briefly checking out what calculus is not

What Calculus Is Not

 No sense delaying the inevitable. Ready for your first calculus test? Circle True or False

 True or False: Unless you actually enjoy wearing a pocket protector, you’ve got no business taking calculus. True or False: Studying calculus is hazardous to your health. True or False: Calculus is totally irrelevant

 False, false, false! There’s this mystique about calculus that it’s this ridiculously difficult, incredibly arcane subject that no one in their right mind would sign up for unless it was a required course.

 Don’t buy into this misconception. Sure, calculus is difficult — I’m not going to lie to you — but it’s manageable, doable. You made it through algebra, geometry, and trigonometry. Well, calculus just picks up where they leave off — it’s simply the next step in a logical progression.

 And calculus is not a dead language like Latin, spoken only by academics. It’s the language of engineers, scientists, and economists. Okay, so it’s a couple steps removed from your everyday life and unlikely to come up at a cocktail party. But the work of those engineers, scientists, and economists has a huge impact on your day-to-day life — from your microwave oven, cell phone, TV, and car to the med icines you take, the workings of the economy, and our national defense. At this very moment, something within your reach or within your view has been impacted by calculus

 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 (assuming 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 calo ries 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 not just in increments like it’s one steepness for the first 3 feet then a different steepness for the next 3 feet. 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 every second or every thousandth of a second, but constantly changing from one moment to the next. That’s what makes it a calculus problem. By this time, it should come as no surprise to you that calculus is described as “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, because it’s straight, 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 the invention of 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 con stantly 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. Here are some more examples.

 With regular math you can determine the length of a buried cable that runs diago nally 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 cir cular arc or a parabola). Knowing the exact length is of obvious importance to a power company planning hundreds of miles of new electric cable. See Figure 1-3

You can calculate the area of the flat roof of a home with ordinary geometry. With calculus you can compute the area of a complicated, nonspherical shape like the dome of the Minneapolis Metrodome. Architects designing such a building 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. Check out Figure 1-4

 With regular math and some simple physics, you can calculate how much a quarterback must lead his receiver to complete a pass. (I’m assuming here that 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 (of different shapes) and the speeds of both are constantly changing — not to mention the fact that on its way to Mars, the spacecraft is affected by the different and constantly changing gravitational pulls of the Earth, the moon, Mars, and the sun. See Figure 1-5.

 You see many real-world applications of calculus throughout this book. The differentiation problems in Part 4 all involve the steepness of a curve — like the steepness of the curving incline in Figure 1-1. In Part 5, you do integration prob lems like the cable-length problem shown back in Figure 1-3. These problems involve breaking up something into little sections, calculating each section, and then adding up the sections to get the total. More about that in Chapter 2.

The Two Big Ideas of Calculus: Differentiation and Integration — plus Infinite Series

This book covers the two main topics in calculus — differentiation and integration — as well as a third topic, infinite series. All three topics touch the earth and the heavens because all are built upon the rules of ordinary algebra and geometry and all involve the idea of infin

Defining Differentiation

 Differentiation is the process of finding the derivative of a curve. And the word “derivative” is just the fancy calculus term for the curve’s slope or steepness. And because the slope of a curve is equivalent to a simple rate (like miles per hour or profit per item), the derivative is a rate as well as a slope

The derivative is a slope

 In algebra, you learned about the slope of a line — it’s equal to the ratio of the rise to the run. In other words, Slope rise run . See Figure 2-1. Let me guess: A sudden rush of algebra nostalgia is flooding over you.

 Just like the line in Figure 2-1, the straight line between A and B in Figure 2-2 has a slope of 1/2. And the slope of this line is the same at every point between A and B. But you can see that, unlike the line, the steepness of the curve is changing between A and B. At A, the curve is less steep than the line, and at B, the curve is steeper than the line. What do you do if you want the exact slope at, say, point C? Can you guess? Time’s up. Answer: You zoom in. See Figure 2-3

 When you zoom in far enough — really far, actually infinitely far — the little piece of the curve becomes straight, and you can figure the slope the old-fashioned way. That’s how differentiation works

The derivative is a rate

 Because the derivative of a curve is the slope — which equals rise run or rise per run — the derivative is also a rate, a this per that like miles per hour orgallons per minute (the name of the particular rate simply depends on the units used on the x- and y-axes). The two graphs in Figure 2-4 show a relationship between distance and time — they could represent a trip in your car

For the problem on the right, on the other hand, you need calculus. (You can’t use the slope formula because you’ve only got one point.) Using the derivative of the curve, you can determine the exact slope or steepness at point C. Just to the left of C on the curve, the slope is slightly lower, and just to the right of C on the curve, the slope is slightly higher. But precisely at C, for a single infinitesimal moment, you get a slope that’s different from the neighboring slopes. The slope for this single infinitesimal point on the curve gives you the instantaneous rate in miles per hour at point C.

 Why Calculus Works

 In Chapters 1 and 2, I talk a lot about the process of zooming in on a curve till it looks straight. The mathematics of calculus works because of this basic nature of curves — that they’re locally straight — in other words, curves are straight at the microscopic level. The earth is round, but to us it looks flat because we’re sort of at the microscopic level when compared to the size of the earth. Calculus works because after you zoom in and curves look straight, you can use regular algebra and geometry with them. The zooming-in process is achieved through the mathematics of limits.

 The Limit Concept: A Mathematical Microscope

 The mathematics of limits is the microscope that zooms in on a curve. Here’s how a limit works. Say you want the exact slope or steepness of the parabola y x the point (1, 1). See Figure 3-1.

With the slope formula from algebra, you can figure the slope of the line between (1, 1) and (2, 4). From (1, 1) to (2, 4), you go over 1 and up 3, so the slope is 3 1 , or just 3. But you can see in Figure 3-1 that this line is steeper than the tangent line at (1, 1) that shows the parabola’s steepness at that specific point. The limit process sort of lets you slide the point that starts at (2, 4) down toward (1, 1) till it’s a thou sandth of an inch away, then a millionth, then a billionth, and so on down to the microscopic level. If you do the math, the slopes between (1, 1) and your moving point would look something like 2.8, then 2.6, then 2.4, and so on, and then, once

you get to a thousandth of an inch away, 2.001, 2.000001, 2.000000001, and so on. And with the almost magical mathematics of limits, you can conclude that the slope at (1, 1) is precisely 2, even though the sliding point never reaches (1, 1). (If it did, you’d only have one point left and you need two separate points to use the slope formula.) The mathematics of limits is all based on this zooming-in process, and it works, again, because the further you zoom in, the straighter the curve gets