The Five College Calculus Course

- James Callahan, Smith College
- David Cox, Amherst College
- Kenneth Hoffman, Hampshire College
- Donal O'Shea, Mount Holyoke College
- Harriet Pollatsek, Mount Holyoke College
- Lester Senechal, Mount Holyoke College

- Calculus in Context (The entire book)
- Calculus I (Chapters 1-6, plus front matter and full index)
- Calculus II (Chapters 7-12, plus front matter and full index)

**Download** individual chapters:

- Prefaces and Table of Contents
- Chapter 1 (A context for calculus: pages 1-60)
- Chapter 2 (Successive approximations: pages 61-100)
- Chapter 3 (The derivative: pages 101-178)
- Chapter 4 (Differential equations: pages 179-272)
- Chapter 5 (Techniques of differentiation: pages 275-336)
- Chapter 6 (The integral: pages 337-418)
- Chapter 7 (Periodicity: pages 419-460)
- Chapter 8 (Dynamical systems: pages 461-510)
- Chapter 9 (Functions of several variables: pages 511-592)
- Chapter 10 (Series and approximations: pages 593-680)
- Chapter 11 (Techniques of integration: pages 681-768)
- Chapter 12 (Case studies: pages 769-834)
- Index (Pages 835-845)

Primary funding for curriculum development and dissemination was provided by the National Science Foundation in grants DMS-14004 (1988-95) and DUE-9153301 (1991-97), awarded to Five Colleges, Inc. Other curriculum development funding has been provided by NECUSE (New England Consortium for Undergraduate Science Education, funded by the Pew Charitable Trusts) to Smith College (1989) and Mount Holyoke College (1990). Five Colleges, Inc. also provided start-up funds.

Equipment and software for computer classrooms has been funded by NSF grants in the ILI program: USE-8951485 to Smith College and DUE/EHR-9551919 to Mount Holyoke College. The Hewlett-Packard Corporation contributed equipment to Mount Holyoke and Smith Colleges, and other equipment was contributed to Mount Holyoke College by IBM and the Sloan Foundation.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.

In this overview, we tell our "creation story" and then describe how it led to the text, spelling out our starting points, our curricular goals, our functional goals, and our view of the impact of technology.

Early in their academic careers, Hampshire students grapple with primary sources in all fields--in economics and ecology, as well as in history and literature. And journal articles don't shelter their readers from home truths: if a mathematical argument is needed, it is used. In this way, students in the life and social sciences found, sometimes to their surprise and dismay, that they needed to know calculus if they were to master their chosen fields. However, the calculus they needed was not, by and large, the calculus that was actually being taught. The journal articles dealt directly with the relation between quantities and their rates of change--in other words, with differential equations.

Confronted with a clear need, those students asked for help. By the
mid-1970s, Michael Sutherland and Kenneth Hoffman were teaching a course
for those students. The core of the course was calculus, but calculus
as it is *used* in contemporary science. Mathematical ideas and
techniques grew out of scientific questions. Given a process, students
had to recast it as a model; most often, the model was a set of
differential equations. To solve the differential equations, they used
numerical methods implemented on a computer.

The course evolved and prospered quietly at Hampshire. More than a
decade passed before several of us at the other four institutions paid
some attention to it. We liked its fundamental premise, that
differential equations belong at the center of calculus. What astounded
us, though, was the revelation that differential equations could really
*be* at the center--thanks to the use of computers.

This book is the result of our efforts to translate the Hampshire course
for a wider audience. The typical student in calculus has not been
driven to study calculus in order to come to grips with his or her own
scientific questions--as those pioneering students had. If calculus is
to emerge organically in the minds of the larger student population, a
way must be found to involve that population in a spectrum of scientific
and mathematical questions. Hence, calculus *in context*.
Moreover, those contexts must be understandable to students with no
special scientific training, and the mathematical issues they raise must
lead to the central ideas of the calculus--to differential equations,
in fact.

Coincidentally, the country turned its attention to the undergraduate science curriculum, and it focused on the calculus course. The National Science Foundation created a program to support calculus curriculum development. To carry out our plans we requested funds for a five-year project; we were fortunate to receive the only multi-year curriculum development grant awarded in the first year of the NSF program. The text and software is the outcome of our effort.

- 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.

- 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.

- 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.

- 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.

increase | decrease | |
---|---|---|

concepts | techniques | |

geometry | algebra | |

graphs | formulas | |

brute force | elegance | |

numerical solutions | 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 $$*y*' = cos^{2}(*t*)). Students also see the
elegant special methods that can be invoked to solve $$*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.

The student population in the first semester course is especially diverse. In fact, since many students take only one semester, the first six chapters stand alone as a reasonably complete course. We have also tried to present the contexts of broadest interest first. The emphasis on the physical sciences increases in the second half of the book.

There are also software programs are available at no charge for use with this text. These are QuickBasic versions of the Basic programs that appear in the text; you can also get QuickBasic itself.

sarah-marie belcastro has produced a collection of notebooks to accompany the text. They are written in both Mathematica and Sage; the latter is a "free open source alternative to Magma, Maple, Mathematica and Matlab."