The meaning of calculus comes first.
A language is best learned when it can be used to make meaningful statements. We begin our book by giving readers a chance to see the power of calculus to describe the physical world. In chapter one students are reading and writing meaningful equations.
Derivatives and integrals are developed side-by-side.
We introduce the symbols, words and concepts for both derivatives and integrals at the beginning, and develop them side-by-side throughout the text. We present all the ideas of calculus quickly and informally, and then slowly sharpen the picture geometrically, numerically, and algebraically. When students have seen the big picture, they are less likely to get lost in the details.
The meaning of calculus is emphasized throughout.
In the early stages, when understanding is only geometric, students can still appreciate the use of mathematical language to explain the ideas of Thomas Malthus and model the growth of a colony. In Chapter 3, when they have some integrating tools, they can plan how to pay back their student loans and investigate an epidemic at the race track.
Throughout the book, we provide contexts for the calculus that are both appropriate and genuine.
The rigorous treatment of limits follows the development of algebraic techniques.
Students need to see limits defined carefully, but they appreciate the subtleties of the definitions and the importance of proofs better after they have some facility with the concepts.
Limits, derivatives, and integrals are defined using sequences. With sequences, it's possible to separate limit arguments into two stages, making definitions considerably simpler than in the standard approach to limits.
Electronic computing devices are incorporated generically.
We take advantage of the insights graphing calculators and computer programs offer, but students are free to use whatever they find most comfortable. Machines enhance, rather than replace mathematics. For example, we use graphers to differentiate with the difference quotient.
Students will find all the tools they need at this website. The tools are easy to use with a minimal amount of class time needed to learn them. The tools include:
Also available here are a few special programs for use in particular assignments.
- Graph- a standard grapher
- Slinky- a simple integrator
- GraphPlus- a surface-drawing grapher for solving differential equations and drawing slope fields, offering many opitions including red-blue 3D pictures
- SlinkyPlus- a more sophisticated integrator, draws vector fields
In addition, there are a series of optional problems in which students use spreadsheets to implement numerical methods and even to create graphics.
Examples from many disciplines offer a choice of course emphasis.
Throughout the book there are sections marked Application, Technique, Theory, and Essay, in which special topics are given a particularly intense or careful treatment.
By choosing which of these sections to include, a wide variety of courses can be constructed: a course for engineers and physicists, a course for pre-business or pre-med students, a course for liberal arts students, a course for mathematics majors, and so on.
There are a number of small touches that make this book unique and engaging for students.
There are a number of laboratories in which students actually gather data then model with differential equations. These range from the physical (The Pendulum) to social (A Rumor Spreads, Weber's Law)
Most exercise sets have a section of ``More Challenging Problems.'' A few have a third section with especially difficult and interesting problems.
Finally, we have exercises that keep on exercising, day after day. We call them Problems Du Jour.
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Calculus I
Calculus II
Calculus III
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Please email your comments and queries to cohenle@smith.edu.