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Introduction

A Preview of Calculus

Calculus was first developed more than three hundred years ago by Sir Isaac Newton and Gottfried Leibniz to help them describe and understand the rules governing the motion of planets and moons. Since then, thousands of other men and women have refined the basic ideas of calculus, developed new techniques to make the calculations easier, and found ways to apply calculus to problems besides planetary motion. Perhaps most importantly, they have used calculus to help understand a wide variety of physical, biological, economic and social phenomena and to describe and solve problems in those areas.

Part of the beauty of calculus is that it is based on a few very simple ideas. Part of the power of calculus is that these simple ideas can help us understand, describe, and solve problems in a variety of fields.

Chapter 1: Review contains review material that you should recall before we begin calculus.

Chapter 2: The Derivative builds on the precalculus idea of the slope of a line to let us find and use rates of change in many situations.

Chapter 3: The Integral builds on the precalculus idea of the area of a rectangle to let us find accumulated change in more complicated and interesting settings.

Chapter 4: Functions of Two Variables extends the calculus ideas of chapter 2 to functions of more than one variable.

Supplements

See MyOpenMath.com for electronic homework. For Grove City College students: announcements, test summary sheets, and other handouts and supplements will be linked from myGCC.

Students who plan to go into science, engineering, or mathematics take a year-long sequence of classes that cover many of the same topics as we do in our one-quarter or one-semester course.

Here are some of the differences:

No trigonometry

We will not be using trigonometry at all in this course. The scientists and engineers need trigonometry frequently, and so a great deal of the engineering calculus course is devoted to trigonometric functions and the situations they can model.

The applications are different

The scientists and engineers learn how to apply calculus to physics problems, such as work. They do a lot of geometric applications, like finding minimum distances, volumes of revolution, or arc-lengths. In this class, we will do only a few of these (distance/velocity problems, areas between curves). On the other hand, we will learn to apply calculus in some economic and business settings, like maximizing profit or minimizing average cost, finding elasticity of demand, or finding the present value of a continuous income stream. These are applications that are seldom seen in a course for engineers.

Fewer theorems, no proofs

The focus of this course is applications rather than theory. In this course, we will use the results of some theorems, but we won't prove any of them. When you finish this course, you should be able to solve many kinds of problems using calculus, but you won't be prepared to go on to higher mathematics.

Less algebra

In this class, you will not need clever algebra. If you need to solve an equation, it will either be relatively simple, or you can use technology to solve it. In most cases, you won't need exact answers; calculator numbers will be good enough.

Simplification and Calculator Numbers

When should you simplify?

• Simplify when it actually makes your life easier. For example, in Chapter 2 it's easier to find a second derivative if you simplify the first derivative.
• Simplify your answer when you need to match it to an answer in the book. You may need to do some algebra to be sure your answer and the book answer are the same.

• When you answer an applied problem, find a calculator number. It doesn’t mean much to suggest that the company should produce $$\dfrac{2.4\cdot\sqrt{12100}}{2.5}$$ items; it’s much more meaningful to report that they should produce about 106 items.