Economics Network CHEER Virtual Edition

Volume 8, Issue 3, 1994

Review Article:
Calculus by Dennis D. Berkey and Paul Blanchard

Barry Murphy
University of Portsmouth

Calculus, by Dennis D. Berkey and Paul Blanchard, third edition, Saunders College Publishing (Harcourt Brace Jovanovich), ISBN 0-03- 075366, 1992, pp. xxxii+ 1150 + 182 (appendices and answers), 18.95

This is the third edition of a successful college text in calculus directed mainly at mathematics, science, and engineering students. Its main features are its great size, huge number of exercises (several thousand), excellent layout (statements of theorems, many worked examples, review exercises and running exercises, introductory historical material, chapter summaries, and highlighted procedural boxes), large amount of supporting material available from the publishers (instructors' manuals, student solutions manuals, test bank in text and computerised form, transparencies, and manuals for calculator and computer enhancement), high production standards (very few typos and largely excellent colour printing), and an orientation towards computer and graphing calculator application. The scale of the publication is therefore enormous, with acknowledgements to several hundred named individuals, and the retail price is therefore astonishingly low. For those who are familiar with the previous edition, the main innovations, apart from a complete overhaul, are the addition of about a thousand new exercises, the direction of material towards computer and graphing calculator implementation, and the provision of three hundred or so exercises (more like mini-projects) for computer and calculator solution. This review will proceed in two parts: first an overall account of the material and its organisation, and then an assessment of the success of the computer orientation.


The core of the book consists of 21 chapters organised into seven units on precalculus, differentiation, integration, transcendental functions, infinite series, plane and solid geometry, and multivariable calculus (including differential equations). The prerequisites are some knowledge of real numbers, coordinate geometry, and trigonometry, though all this is well reviewed both in the introductory chapter and periodically. The coverage of the book is similar to that of the first volume of Tom M. Apostol's two-volume Calculus (Wiley, 1969), though at a lower level of abstraction; the level of treatment is similar to that of Binmore's Calculus (Cambridge 1983), though the coverage and orientation of the latter are substantially different.

Despite the focus on computer application, and a wonderfully-illustrated "fractal" appendix on the convergence properties of Newton's method, the range of material is surprisingly traditional, covering the main topics of nineteenth-century calculus in a non-analytical applications-oriented way. Limits, differentiation, antidifferentiation, extremisation, the trigonometric and hyperbolic functions, the definite integral, solids and surfaces of revolution, the transcendental functions, infinite series, Taylor series and power series, line, surface, and multiple integrals, and vector analysis up to Stokes' theorem are all covered in an easy, carefully considered progression.

While many results are proved, the almost invariable approach is to state and illustrate a theorem with many worked examples, deferring proof to a later stage of the text or to an exercise or to an appendix at the end of the book; many results are never established, though repeatedly applied (e.g. the intermediate value theorem and Weierstrass' theorem on continuous functions). This is not intended to be a criticism but rather a description of the approach: the focus is on giving the student "good reason" for believing in the results and methods stated, and an ability to apply them. It would take a very clear-minded reader indeed to be able to establish the logical dependence of any given result (say, the mean value theorem) on earlier material proved or quoted.

There have to be some reservations. With such a large number of problems, some are trivial or repetitive, and the occasional examples and applications to statistics and to economics are rather disappointing; probability is evidently regarded as a branch of "finite mathematics". A non-economist reading the text could gain the impression that economics is either a trivial or essentially non-quantitative subject; and a professional economist would be astonished to read (twice) that the magnitude of a Lagrange multiplier is of no significance (an untypical lapse). The coverage of multivariable calculus is also unsuitable for the mathematical specialist and for applications in economics and statistics: this is very definitely "vector analysis" with applications to engineering and physics, and not the theory of linear spaces.

That said, these are very minor reservations. One is astonished throughout at the care given to presentation, and the excellence of the worked examples and the many exercises. A very successful calculus course for mathematicians, engineers, physicists, and (with judicious selection by the lecturer) economists, can be based on the book. The repeated application of the method of Riemann sums to a huge variety of problems is particularly successful. The very best thing that can be said about the book is that, by virtue of its layout and worked examples, it is quite impossible to read without learning a huge amount of mathematics and applied mathematics, and this is to say a lot since the book is very readable indeed.

Computer and Graphing Calculator Orientation

One can think of at least three ways in which a computer or graphing calculator (from now on usually "computer" for short) can be used in teaching mathematics:

The book makes no claim to address the first point: computer specialists will eventually need a further and quite different course. The second point is well implemented: using Wolfram Research Inc's Mathematica 2.0 as a basis, a huge number of high-quality two- and three- dimensional diagrams is presented. There are some reservations to be made here: (a) there have been many excellently illustrated mathematics texts developed before the era of computer graphics though the latter allows a profusion and use of colour that was not previously available; (b) though the colour illustrations are on the whole excellent and well-illuminated, there are occasional problems with illustrations (say of discontinuous functions of two variables), though this may be more a matter of production rather than design; (c) there is still room for much further development: as an example, while quite a lot of emphasis is placed on the use of contour plots and surface plots in two- and three- variable differentiation, these devices are by no means fully integrated with one another, and one must expect improvements in future editions.

My remaining remarks relate to the third point, that of the use of the computer as a learning aid. It may be useful to enumerate the materials provided:

  1. three hundred or so problems for solution on the computer or graphing calculator;

  2. a 38-page appendix contributed by James D. Angelos on calculus and the graphing calculator;

  3. a 28-page appendix by the authors on calculus and the computer, giving 9 equivalent short programs in BASIC and Pascal, and a small range of simple Mathematica applications;

  4. a 10-page appendix illustrating the convergence behaviour of Newton's method for a problem in the complex plane;

  5. a range of manuals (not supplied with the core text) for calculus on the calculator, calculus and Mathematica, calculus and Derive (the Soft Warehouse product), and other utilities.

Rather than review these materials in detail I make some general points. First, the calculator exercises are very largely successful, and seem easy to implement in a variety of ways: I have tried a range of them on calculator, Mathematica, Derive Version 2, Mathcad Version 3.1 (the Adept Scientific product), and on spreadsheet. These problems are more in the nature of mini-projects and any one make take students several hours to solve: the authors warn instructors against assigning a problem before implementing its solution themselves. Next, because of the range of calculators and software utilities available, the materials provide only a skeletal introduction: the student will have to learn BASIC or Mathematica or whatever, and develop his own routines using the indications provided. It has also to be remarked that the wonderful illustrations of Newton's method will be beyond the range of almost all students and many lecturers: highly specialised graphical software and hardware is a prerequisite, even for monochrome implementation; it seems fair to say, therefore that this appendix is not integrated into the text and is more of a styling device to generate a pretty picture for the cover of the book.

I finish by noting that, possibly because of the lack of orientation towards students of business and the social sciences, a big opportunity has been missed: spreadsheets are simply not mentioned. As far as I can see, almost all the calculator exercises presented can be quite easily (in my view more easily) implemented on widely available spreadsheets such as Lotus 1-2-3 or Quattro Pro, with relatively little ingenuity and without recourse even to macros (unless more than a "one-off" solution is required. The only reservations are that analytical solutions have to be provided by the student (little harm anyway, and this also applies to BASIC, Pascal, and many calculators), that a spreadsheet is unsuitable for graphical presentation of higher dimensional problems (the same comment applies), and that certain trigonometric and hyperbolic functions may have to be supplied (e.g. sec = 1/cos). To establish my point I present two self-explanatory applications in Boxes 1 and 2 and Figures 1 and 2. The first application took about 30 minutes to implement and test, and the second about 90 minutes.

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