# Calculus in Context - 2008 Edition

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James Callahan, Northampton, Massachusetts

David Cox, Amherst, Massachusetts

Kenneth Hoffman, Amherst, Massachusetts

Language: English

## Conditions of Use

CC BY

• 1. A Context for Calculus
• 2. Successive Approximations
• 3. The Derivative
• 4. Differential Equations
• 5. Techniques of Differentiation
• 6. The Integral
• 7. Periodicity
• 8. Dynamical Systems
• 9. Functions of Several Variables
• 10. Series and Approximations
• 11. Techniques of Integration
• 12. Case Studies

## Ancillary Material

• Smith College Open Educational Resources: Textbooks

#### Designing the curriculum

We believe that calculus can be for students what it was for Euler and the Bernoullis: a language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in connections to other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines. We began our work with a "clean slate," not by asking what parts of the traditional course to include or discard. Our starting points are thus our summary of what calculus is really about. Our curricular goals are what we aim to convey about the subject in the course. Our functional goals describe the attitudes and behaviors we hope our students will adopt in using calculus to approach scientific and mathematical questions.

#### Starting Points

• Calculus is fundamentally a way of dealing with functional relationships that occur in scientific and mathematical contexts. The techniques of calculus must be subordinate to an overall view of the questions that give rise to these relationships.
• Technology radically enlarges the range of questions we can explore and the ways we can answer them. Computers and graphing calculators are much more than tools for teaching the traditional calculus.
• The concept of a dynamical system is central to science. Therefore, differential equations belong at the center of calculus, and technology makes this possible at the introductory level.
• The process of successive approximation is a key tool of calculus, even when the outcome of the process--the limit--cannot be explicitly given in closed form.

#### Curricular Goals

• Develop calculus in the context of scientific and mathematical questions.
• Treat systems of differential equations as fundamental objects of study.
• Construct and analyze mathematical models.
• Use the method of successive approximations to define and solve problems.
• Develop geometric visualization with hand-drawn and computer graphics.
• Give numerical methods a more central role.

#### Functional Goals

• Encourage collaborative work.
• Enable students to use calculus as a language and a tool.
• Make students comfortable tackling large, messy, ill-defined problems.
• Foster an experimental attitude towards mathematics.
• Help students appreciate the value of approximate solutions.
• Teach students that understanding grows out of working on problems.

#### Impact of Technology

• Differential equations can now be solved numerically, so they can take their rightful place in the introductory calculus course.
• The ability to handle data and perform many computations makes exploring messy, real-world problems possible.
• Since we can now deal with credible models, the role of modelling becomes much more central to the subject.

The text illustrates how we have pursued the curricular goals. Each goal is addressed within the first chapter which begins with questions about describing and analyzing the spread of a contagious disease. A model is built: a model which is actually a system of coupled non-linear differential equations. We then begin a numerical exploration on those equations, and the door is opened to a solution by successive approximations. Our implementation of the functional goals is also evident. The text has many more words than the traditional calculus book--it is a book to be read. The exercises make unusual demands on students. Most are not just variants of examples that have been worked in the text. In fact, the text has rather few "template'' examples.

#### Shifts in Emphasis

It will also become apparent to you that the text reflects substantial shifts in emphasis in comparison to the traditional course. Here are some of the most striking:

How the emphasis shifts:

increase: concepts, geometry, graphs, brute force, numerical solutions

decrease: techniques, algebra, formulas, elegance, closed-form solutions

Since we all value elegance, let us explain what we mean by "brute force." Euler's method is a good example. It is a general method of wide applicability. Of course when we use it to solve a differential equation like y'(t) = t, we are using a sledgehammer to crack a peanut. But at least the sledgehammer works. Moreover, it works with coconuts (like y' = y(1 - y/10)), and it will even knock down a house (like y' = cos2(t)). Students also see the elegant special methods that can be invoked to solve y' = t and y' = y(1 - y/10) (separation of variables and partial fractions are discussed in chapter 11), but they understand that they are fortunate indeed when a real problem will succumb to such methods.