Calculus For Dummies

Introduction

The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have this notion that calculus is impossibly difficult unless you happen to be a direct descendant of Einstein.

Well, I’m here to tell you that you can master calculus. It’s not nearly as tough as its mystique would lead you to think. Much of calculus is really just very advanced algebra, geometry, and trig. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.

But why should you bother — apart from being required to take a course? Why climb Mt. Everest? Why listen to Beethoven’s Ninth Symphony? Why visit the Louvre to see the Mona Lisa? Why watch South Park? Like these endeavors, doing calculus can be its own reward. There are many who say that calculus is one of the crowning achievements in all of intellectual history. As such, it’s worth the effort. Read this jargon-free book, get a handle on calculus, and join the happy few who can proudly say, Calculus? Oh, sure, I know calculus. It’s no big deal.

About This Book

Calculus For Dummies, 2nd Edition, is intended for three groups of readers: students taking their first calculus course, students who need to brush up on their calculus to prepare for other studies, and adults of all ages who’d like a good introduction to the subject.

If you’re enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.

If you’ve had elementary calculus, but it’s been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies, 2nd Edition will give you a thorough, no-nonsense refresher course.

Non-student readers will find the book’s exposition clear and accessible. Calculus For Dummies, 2nd Edition, takes calculus out of the ivory tower and brings it down to earth.

This is a user-friendly math book. Whenever possible, I explain the calculus concepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. I then show you how the calculus concepts work in concrete examples. Only later do I give you the fancy calculus formulas. All explanations are in plain English, not math-speak.

The following conventions keep the text consistent and oh-so-easy to follow:

Variables are in italics.

Calculus terms are italicized and defined when they first appear in the text.

In the step-by-step problem-solving methods, the general action you need to take is in bold, followed by the specifics of the particular problem.

It can be a great aid to true understanding of calculus — or any math topic for that matter — to focus on the why in addition to the how-to. With this in mind, I’ve put a lot of effort into explaining the underlying logic of many of the ideas in this book. If you want to give your study of calculus a solid foundation, you should read these explanations. But if you’re really in a hurry, you can cut to the chase and read only the important introductory stuff, the example problems, the step-by-step solutions, and all the rules and definitions next to the icons. You can then read the remaining exposition only if you feel the need.

I find the sidebars interesting and entertaining. (What do you expect? I wrote them!) But you can skip them without missing any essential calculus. No, you won’t be tested on this stuff.

Minor note: Within this book, you may note that some web addresses break across two lines of text. If you’re reading this book in print and want to visit one of these web pages, simply key in the web address exactly as it’s noted in the text, as though the line break doesn’t exist. If you’re reading this as an e-book, you’ve got it easy — just click the web address to be taken directly to the web page.

Foolish Assumptions

Call me crazy, but I assume …

You know at least the basics of algebra, geometry, and trig.

If you’re rusty, Part 2 (and the online Cheat Sheet) contains a good review of these pre-calculus topics. Actually, if you’re not currently taking a calculus course, and you’re reading this book just to satisfy a general curiosity about calculus, you can get a good conceptual picture of the subject without the nitty-gritty details of algebra, geometry, and trig. But you won’t, in that case, be able to follow all the problem solutions. In short, without the pre-calculus stuff, you can see the calculus forest, but not the trees. If you’re enrolled in a calculus course, you’ve got no choice — you’ve got to know the trees as well as the forest.

You’re willing to do some w_ _ _ .

No, not the dreaded w-word! Yes, that’s w-o-r-k, work. I’ve tried to make this material as accessible as possible, but it is calculus after all. You can’t learn calculus by just listening to a tape in your car or taking a pill — not yet anyway.

Is that too much to ask?

Icons Used in This Book

Keep your eyes on the icons:

mathrules Next to this icon are calculus rules, definitions, and formulas.

remember These are things you need to know from algebra, geometry, or trig, or things you should recall from earlier in the book.

tip The bull’s-eye icon appears next to things that will make your life easier. Take note.

warning This icon highlights common calculus mistakes. Take heed.

Beyond the Book

There’s some great supplementary calculus material online that you might want to check out:

To view this book’s Cheat Sheet, simply go to www.dummies.com and search for Calculus For Dummies Cheat Sheet in the Search box; you’ll find a nice list of important formulas, theorems, definitions, and so on from algebra, geometry, trigonometry, and calculus. This is a great place to go if you forget a formula.

At www.dummies.com, there are articles on some calculus topics that many calculus courses skip. For example, the online article Finding Volume with the Matryoshka Doll Method (a.k.a. the Cylindrical Shell Method) covers one of the methods for computing volume that used to be part of the standard calculus curriculum, but which is now often omitted. You’ll also find other interesting, off-the-beaten-path calculus articles. Check them out if you just can’t get enough calculus.

Where to Go from Here

Why, Chapter 1, of course, if you want to start at the beginning. If you already have some background in calculus or just need a refresher course in one area or another, then feel free to skip around. Use the table of contents and index to find what you’re looking for. If all goes well, in a half a year or so, you’ll be able to check calculus off your list:

___ Run a marathon

___ Go skydiving

___ Write a book

Learn calculus

___ Swim the English Channel

___ Cure cancer

___ Write a symphony

___Pull an inverted 720° at the X-Games

For the rest of your list, you’re on your own.

Part 1

An Overview of Calculus

IN THIS PART …

A brief and straightforward explanation of just what calculus is. Hint: It’s got a lot to do with curves and with things that are constantly changing.

Examples of where you might see calculus at work in the real world: curving cables, curving domes, and the curving path of a spacecraft.

The first of the two big ideas in calculus: differentiation, which means finding a derivative. A derivative is basically just the fancy calculus version of a slope; and it’s a simple rate — a this per that.

The second big calculus idea: integration. It’s the fancy calculus version of adding up small parts of something to get the total.

An honest-to-goodness explanation of why calculus works: In short, it’s because when you zoom in on curves (infinitely far), they become straight.

Chapter 1

What Is Calculus?

IN THIS CHAPTER

You’re only in Chapter 1 and you’re already going to get your first calc test

Calculus — it’s just souped-up regular math

Zooming in is the key

The world before and after calculus

My best day in Calc 101 at Southern Cal was the day I had to cut class to get a root canal.

— MARY JOHNSON

I keep having this recurring dream where my calculus professor is coming after me with an axe.

— TOM FRANKLIN, COLORADO COLLEGE SOPHOMORE

Calculus is fun, and it’s so easy. I don’t get what all the fuss is about.

— SAM EINSTEIN, ALBERT’S GREAT-GRANDSON

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 medicines 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).

FIGURE 1-1: The difference between regular math and calculus: In a word, it’s the curve.

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 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.

FIGURE 1-2: Zooming in on the curve — voilà, it’s straight (almost).

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 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. Here are some more examples.

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. See Figure 1-3.

FIGURE 1-3: Without and with calculus.

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.

FIGURE 1-4: Sans and avec calculus.

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.

FIGURE 1-5: B.C.E. (Before the Calculus Era) and C.E. (the Calculus Era).

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 problems 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.

Chapter 2

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

IN THIS CHAPTER

Delving into the derivative: It’s a rate and a slope

Investigating the integral — addition for experts

Infinite series: Achilles versus the tortoise — place your bets

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 infinity.

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, . See Figure 2-1. Let me guess: A sudden rush of algebra nostalgia is flooding over you.

FIGURE 2-1: The slope of a line equals the rise over the run.

In Figure 2-1, the rise is half as long as the run, so the line has a slope of 1/2.

On a curve, the slope is constantly changing, so you need calculus to determine its slope. See Figure 2-2.

FIGURE 2-2: The slope of a curve isn’t so simple.

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.

FIGURE 2-3: Zooming in on the curve.

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 or rise per run — the derivative is also a rate, a this per that like miles per hour or gallons 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.

FIGURE 2-4: Average rate and instantaneous rate.

A regular algebra problem is shown on the left in Figure 2-4. If you know the x- and y-coordinates of points A and B, you can use the slope formula to calculate the slope between A and B, and, in this problem, that slope gives you the average rate in miles per hour for the interval from A to B.

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.

Investigating Integration

Integration is the second big idea in calculus, and it’s basically just fancy addition. Integration is the process of cutting up an area into tiny sections, figuring the areas of the small sections, and then adding up the little bits of area to get the whole area. Figure 2-5 shows two area problems — one that you can do with geometry and one where you need calculus.

FIGURE 2-5: If you can’t determine the area on the left, hang up your calculator.

The shaded area on the left is a simple rectangle, so its area, of course, equals length times width. But you can’t figure the area on the right with regular geometry because there’s no area formula for this funny shape. So what do you do? Why, zoom in, of course. Figure 2-6 shows the top portion of a narrow strip of the weird shape blown up to several times its size.

FIGURE 2-6: For the umpteenth time, when you zoom in, the curve becomes straight.

When you zoom in as shown in Figure 2-6, the curve becomes practically straight, and the further you zoom in, the straighter it gets. After zooming in, you get the shape on the right in Figure 2-6, which is practically an ordinary trapezoid (its top is still slightly curved). Well, with the magic of integration, you zoom in infinitely close (sort of — you can’t really get infinitely close, right?). At that point, the shape is exactly an ordinary trapezoid — or, if you want to get really basic, it’s a triangle sitting on top of a rectangle. Because you can compute the areas of rectangles, triangles, and trapezoids with ordinary geometry, you can get the area of this and all the other thin strips and then add up all these areas to get the total area. That’s integration.

Figure 2-7 has two graphs of a city’s electrical energy consumption on a typical summer day. The horizontal axes show the number of hours after midnight, and the vertical axes show the amount of power (in kilowatts) used by the city at different times during the day.

FIGURE 2-7: Total kilowatt-hours of energy used by a city during a single day.

The crooked line on the left and the curve on the right show how the number of kilowatts of power depends on the time of day. In both cases, the shaded area gives the number of kilowatt-hours of energy consumed during a typical 24-hour period. The shaded area in the oversimplified and unrealistic problem on the left can be calculated with regular geometry. But the true relationship between the amount of power used and the time of day is more complicated than a crooked straight line. In a realistic energy-consumption problem, you’d get something like the graph on the right. Because of its weird curve, you need calculus to determine the shaded area. In the real world, the relationship between different variables is rarely as simple as a straight-line graph. That’s what makes calculus so useful.

Sorting Out Infinite Series

Infinite series deal with the adding up of an infinite number of numbers. Don’t try this on your calculator unless you’ve got a lot of extra time on your hands. Here’s a simple example. The following sequence of numbers is generated by a simple doubling process — each term is twice the one before it:

1, 2, 4, 8, 16, 32, 64, 128, …

The infinite series associated with this sequence of numbers is just the sum of the numbers:

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + …

Divergent series

The preceding series of doubling numbers is divergent because if you continue the addition indefinitely, the sum will grow bigger and bigger without limit. And if you could add up all the numbers in this series — that’s all infinitely many of them — the sum would be infinity. Divergent usually means — there are exceptions — that the series adds up to infinity.

Divergent series are rather uninteresting because they do what you expect. You keep adding more numbers, so the sum keeps growing, and if you continue this forever, the sum grows to infinity. Big surprise.

Convergent series

Convergent series are much more interesting. With a convergent series, you also keep adding more numbers, the sum keeps growing, but even though you add numbers forever and the sum grows forever, the sum of all the infinitely many terms is a finite number. This surprising result brings me to Zeno’s famous paradox of Achilles and the tortoise. (That’s Zeno of Elea, of course, from the 5th century B.c.)

Achilles is racing a tortoise — some gutsy warrior, eh? Our generous hero gives the tortoise a 100-yard head start. Achilles runs at 20 mph; the tortoise runs at 2 mph. Zeno used the following argument to prove that Achilles will never catch or pass the tortoise. If you’re persuaded by this proof, by the way, you’ve really got to get out more.

Imagine that you’re a journalist covering the race for Spartan Sports Weekly, and you’re taking a series of photos for your article. Figure 2-8 shows the situation at the start of the race and your first two photos.