Page 19 - Calculus Workbook For Dummies

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Introduction

Part II: Limits and Continuity

You can actually do most practical calculus problems without knowing much about
limits and continuity. The calculus done by scientists, engineers, and economists
involves differential and integral calculus (see Parts III and IV), not limits and continu-
ity. But because mathematicians do care about limits and continuity and because
they’re the ones who write calculus texts and design calculus curricula, you have to
learn these topics.

Obviously, I’m being a bit cynical here. Limits and continuity are sort of the logical
scaffolding that holds calculus up, and, as such, they’re topics worthy of your time
and effort.

Part III: Differentiation

Differentiation and integration (Part IV) are the two big ideas in calculus. Differentiation
is the study of the derivative, or slope, of functions: where the slope is positive, nega-
tive, or zero; where the slope has a minimum or maximum value; whether the slope
is increasing or decreasing; how the slope of one function is related to the slope of
another; and so on. In Part III, you get differentiation basics, differentiation rules, and
techniques for analyzing the shape of curves, and solving problems with the derivative.

Part IV: Integration and Infinite Series

Like differentiation, “integration” is a fancy word for a simple idea: addition. Every
integration problem involves addition in one way or another. What makes integration
such a big deal is that it enables you to add up an infinite number of infinitely small
amounts. Using the magic of limits, integration cuts up something (an area, a volume,
the pressure on the wall of a tank, and so on) into infinitely small chunks and then
adds up the chunks to arrive at the total. In Part IV, you work through integration
basics, techniques for finding integrals, and problem solving with integration.

Infinite series is a fascinating topic full of bizarre, counter-intuitive results, like the infi-
nitely long trumpet shape that has an infinite surface area but a finite volume! — hard
to believe but true. Your task with infinite series problems is to decide whether the sum
of an infinitely long list of numbers adds up to infinity (something that’s easy to imag-
ine) or to some ordinary, finite number (something many people find hard to imagine).

Part V: The Part of Tens

Here you get ten things you should know about limits and infinite series, ten things
you should know about differentiation, and ten things you should know about integra-
tion. If you find yourself knowing no calculus with your calc final coming up in 24
hours (perhaps because you were listening to Marilyn Manson on your iPod during
class and did all your assignments in a “study” group), turn to the Part of Tens and the
Cheat Sheet. If you learn only this material — not an approach I’d recommend — you
may actually be able to barely survive your exam.