It is good to see that in the midst of the oppressive sameness that characterizes the world of calculus textbooks there is still some space for new approaches. Cohen and Henle, both professors at Smith College, have had the courage to write a book that is not like the others, and the result is quite good, and probably pedagogically effective in the right kind of classroom.
One could describe the approach as analogous to natural language learning: first one learns to speak simple sentences, then to understand meanings more deeply, then to construct more complex sentences. Only after that is it a good idea to study grammar. So Cohen and Henle introduce the idea of derivatives and integrals in chapter one: derivatives are rates of change, integrals are the overall accumulated change given the rate. Then come the pictures: graphs, slopes, the relationship between the graph of the function and the graph of the derivative (in both directions). Next, approximations: difference quotients and something like Riemann sums. (It is only at this point that the connection between integrals and area comes up.) Then, algebraic methods, the rules for derivatives and basic integration. And finally a chapter attempting to lay down the theory behind all this, using a sequential approach to the definitions of limits and continuity.
That only gets us halfway through the book, but it should be enough to make it clear that the authors are not afraid to be different. The sections that follow continue in that style. While they hit on all the high points that one would expect, including some multivariable calculus and some simple complex variables material, they do it in their own way, which is often quite insightful and creative.
Throughout the book are sprinkled many applications, most of which boil down to differential equations. The authors here are (silently) following the historical roots of the subject: the calculus was first of all the calculus of differentials, and its main point from the beginning was the solution of differential equations. They also follow the historical path in privileging the intuitive over the formal. This doesn't mean that the book is less than rigorous; the ideas are there. It's just that they aren't formalized into the fancy dress of "analysis" until after they have been appropriated in other ways. In fact, a final chapter includes full proofs of such things as the Intermediate Value Theorem, which aren't to be found at all in most calculus books. So if the path is followed all the way to the end the result is actually more rigorous than your typical calculus text.
One worry I have has to do with the changing demographics of calculus courses. In a class full of students who have taken a calculus course in high school, the professor would probably find it very hard to "sell" this approach. The careful build-up of concepts that the book proposes would likely run into a lot of resistance from the "nxn-1" crowd. By contrast, using this book with students who have not met these concepts before (perhaps even in a high school course... perhaps especially in a high school course) would probably work very well.
I should add a note saluting the author's courage in using d where most of us would follow tradition and use ε. I wouldn't have dared.
Overall, I'm impressed. If I were assigned to teach calculus in high school, this is probably the text I'd try. But even in a college, with students who are starting out, this is a book I'd like to try out, just to see if it can be made to work. I think there's a good chance it could.
Fernando Q. Gouvêa is professor of mathematics at Colby College, where he has taught second-semester calculus more times than he can count.
Calculus: The Language of Change
David W. Cohen, Smith College, James M. Henle, Smith College
Chapter 1. Introducing the Words
Chapter 2. Introducing the Pictures
Chapter 3. Approximating Derivatives and Integrals
Chapter 4. Algebraic Methods
Chapter 5. The Basic Theory
Chapter 6. Modeling Tools
Chapter 7. Aspects of Integration
Chapter 8. Polynomial Approximations
Chapter 9. Infinite Series
Chapter 10. Surfaces
Chapter 11. Complex Calculus
Chapter 12. The Deeper Theory
Appendix A: A Review of Algebra and Geometry
Appendix B: A Review of Trigonometry
Appendix C: A Table of Integrals
Appendix D: Answers, Notes, and Hints