*by Steve Kennedy, MAA Senior Acquisitions Editor*

What do you do at your school with a first–year student who scored a four or five in AP calculus in high school? There is, at least at Carleton, an awkward articulation between the coverage of the AP AB syllabus and our first-year calculus sequence.

For the first several weeks of our Calculus II class we are covering material that is, at least partially, in an AP calculus class: methods of integration; applications of the integral, including arc length and surface area; some elementary differential equations. But students who take Calculus I at Carleton haven’t seen any of this. It makes for a strange dynamic in the classroom: For some students everything is new and confusing; for others (the AP students) large parts are review. The former feel at a disadvantage, the latter a little bored.

We don’t really want to have two strands through our calculus sequence, one for students with AP experience and another for students who take all their calculus at Carleton. Several years ago we took the AP-experienced students and gave them a half-term course in modeling while the non-AP students took our standard Calculus II material. Then the two groups came together for the second half of the term to study sequences and series together. This turned out to be logistically difficult (and confusing to folks in other departments of the college), and we abandoned the experiment after a few years.

So, I was prepared to be interested when Michael Boardman and Roger Nelsen produced

*College Calculus: A One-Term Course for Students with Previous Calculus Experience*. Michael and Roger have, together, more than 40 years’ experience grading, writing, and consulting on the AP calculus exam. They are as familiar with what a student who scores a four or a five on that exam knows as it is possible to be. Their take on the issue was bound to be worthwhile.

First, the book material includes all the standard material in any second-semester calculus class: methods of integration; applications of integration including arc length, surface area, volume, work and center of mass; differential equations; hyperbolic functions; numerical integration; parametric curves and polar coordinates; improper integrals; and infinite series.

But the list of topics doesn’t do this book justice. I love what the authors have chosen to do with the (usually deadly boring) collection of integration techniques. There is no chapter of techniques; they are sprinkled throughout the text and produced as needed. So, integration by parts comes up in the volume chapter; trig substitutions are introduced in the chapter on arc length; partial fractions come to rescue when the logistic growth ODE leaves the reader stumped. It’s a clever idea and makes this part of the course much more palatable and relevant.

The other pedagogical innovation that distinguishes this text is the Explorations section that concludes each chapter. These are collections of meaty, worthwhile exploratory problems. For example, the first chapter has the reader, inter alia, deriving a formula for the antiderivative of the product of an exponential and polynomial and proving the Cauchy-Schwartz inequality for integrals. The chapter on numerical integration has explorations that cleverly exploit symmetries to integrate otherwise intractable integrands and demonstrates the Hermite-Hadamard inequality.

By the way, the numerical integration chapter has a very slick proof that Simpson’s rule is exact for cubics. To wit, look at three consecutive points of subdivision, suppose ƒ is a cubic and

*g*is the quadratic that agrees with ƒ on these three points, consider ƒ-

*g*: This has a lovely symmetry across the middle. Notice it is a cubic and you know all three zeroes, the left- and right-most zeroes are symmetrically placed with respect to the middle zero, which means this cubic integrates to zero across this interval. (Draw a picture.)

This book is full of beautiful things like that. The authors have made a conscious effort to raise the level of discourse above that of a typical calculus textbook. They are addressing students who already have a degree of mastery of some elements of calculus, and you can see it in their exposition. Everything is crystal clear and beautifully explained, but not in a dumbed-down or pandering way. Your students will be able to read this text, and they should want to.

Approximately 300,000 high school students will take the AP AB exam this spring. About 40 percent of them will earn a four or a five. What are we going to do with these very-well-prepared folks when they get to our calculus classes next year? Most of them would be very well served by this well-written and thoughtfully constructed book.

This article was written for MAA FOCUS as part of MAA Books Beat. It appears in the April/May 2015 issue.

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