Calculus For Dummies

Mark Ryan

• Editore: For Dummies
• Anno: 2016
• ISBN:
  9781119297437
• Edizione: 2
• Argomento: Matemáticas
• Categoria: Cálculo

Slay the calculus monster with this user-friendly guide

Calculus For Dummies, 2nd Edition makes calculus manageable—even if you're one of the many students who sweat at the thought of it. By breaking down differentiation and integration into digestible concepts, this guide helps you build a stronger foundation with a solid understanding of the big ideas at work. This user-friendly math book leads you step-by-step through each concept, operation, and solution, explaining the "how" and "why" in plain English instead of math-speak. Through relevant instruction and practical examples, you'll soon learn that real-life calculus isn't nearly the monster it's made out to be.

Calculus is a required course for many college majors, and for students without a strong math foundation, it can be a real barrier to graduation. Breaking that barrier down means recognizing calculus for what it is—simply a tool for studying the ways in which variables interact. It's the logical extension of the algebra, geometry, and trigonometry you've already taken, and Calculus For Dummies, 2nd Edition proves that if you can master those classes, you can tackle calculus and win.

• Includes foundations in algebra, trigonometry, and pre-calculus concepts
• Explores sequences, series, and graphing common functions
• Instructs you how to approximate area with integration
• Features things to remember, things to forget, and things you can't get away with

Stop fearing calculus, and learn to embrace the challenge. With this comprehensive study guide, you'll gain the skills and confidence that make all the difference. Calculus For Dummies, 2nd Edition provides a roadmap for success, and the backup you need to get there.

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 eno...