Calculus for Dummies

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Calculus For Dummies

John Wiley & Sons

Chapter One

* * *

In This Chapter

* You're only on page 1 and you've got a calc test already

* 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 theinevitable. Ready for your first calculus test? Answer
True or False.

T F Unless you actually enjoy wearing a pocket protector, you've got no
business taking calculus.

T F Studying calculus is hazardous to your health.

T F 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 is
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).

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 10 feet then a different steepness for the next 10 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 lets regular math finish off the problem.

What makes calculus such a fantastic achievement is that it actually zooms in
infinitely. In fact, everything you do 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. 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.

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 Houston Astrodome. 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 by how
much a quarterback must lead his receiver to complete a pass. Note 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 IV all involve the steepness of a curve - like
the steepness of the curving incline in Figure 1-1. In Part V, 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 this in
Chapter 2.

(Continues...)

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Excerpted from Calculus For Dummies
by Mark Ryan
Copyright © 2003 by Mark Ryan.
Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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