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.
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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:
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 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.
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.
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.
When you were in tenth grade, your math teacher may have impressed you with the need to simplify your answers. I'm here to tell you – she was wrong. The form your answer should be in depends entirely on what you will do with it next. In addition, the process of simplifying , often messy algebra, can ruin perfectly correct answers. From the teacher’s point of view, simplifying obscures how a student arrived at his answer, and makes problems harder to grade. Moral: don't spend a lot of extra time simplifying your answer. Leave it as close to how you arrived at it as possible.
A calculator is required for this course, and it can be a wonderful tool. However, you should be careful not to rely too strongly on your calculator. Follow these rules of thumb:
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Business Calculus lecture videos by Eric Bancroft are licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The main body of the text is licensed under a Creative Commons Attribution 4.0 International License.
This text was originally adapted from Business Calculus (and the later Applied Calculus) by Shana Calaway, Dale Hoffman, and David Lippman. What follows is the original license from that text.
Copyright © 2013 Shana Calaway, Dale Hoffman, David Lippman.
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Chapter 1 was remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. It was adapted for this text by David Lippman, and is used under the Creative Commons Attribution license by permission of the authors.
Chapters 2-4 were created by Shana Calaway, remixed from Contemporary Calculus by Dale Hoffman, and edited and extended by David Lippman.
Shana Calaway teaches mathematics at Shoreline Community College.
Dale Hoffman teaches mathematics at Bellevue College. He is the author of the open textbook Contemporary Calculus.
David Lippman teaches mathematics at Pierce College Ft Steilacoom. He is the coauthor of the open textbooks Precalculus: An Investigation of Functions and Math in Society.