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Calculus Problems

Marco Baronti, Filippo De Mari, Robertus van der Putten, and Irene Venturi
Publication Date: 
Number of Pages: 
Unitext 101
Problem Book
[Reviewed by
Allen Stenger
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This is a collection of calculus problems, organized into short chapters by subject area. Each chapter starts with a set of definitions and theorems, then some “guided exercises” (worked problems), and a large collection of exercises, with brief answers in the back.

So far this sounds a lot like Schaum’s Outlines: Calculus, and in fact there are strong similarities between the books, although the present book only covers single-variable calculus, series, and ordinary differential equations while Schaum also does multi-variable. The present book is more talkative and has much more difficult functions in the examples and exercises (which are therefore probably more realistic). In both books nearly all the exercises concern properties of specific stated functions rather than general theorems, although the present book poses some more open-ended problems (usually near the end of the chapter) and has just a few proof problems.

Overall I rate the raw difficulty of the two books about the same, although the present book is not compartmentalized as much as Schaum (or most courses) and requires you to remember things from other areas of calculus. For example, the series problems in the present book often involve logs, exponentials, and trig functions, so you really have to have a good understanding of these to solve the problems. The last chapter is miscellaneous problems that are given without any hints to the solution methods to use, and so would be very challenging for most students.

A couple of samples of the more advanced topics: (1) on p. 62ff are several examples of finding the limit of a sequence defined by a recursion; for example, given \(a_1 = 2\) and \(a_{n+1} = (a_n^2 + 3)/(2a_n)\), show \(\lim a_n\) exists and find it. These usually require several stages of bootstrapping (although the book does not use this term), where we successively show the sequence is positive, then monotonic (and so goes to a limit), and then find the limit by taking the limit of the recursion. (2) One of the few proofs: On p. 232 the beautiful but little-known theorem that if a sum \(\sum a_n\) of monotone decreasing positive terms converges, then \(n a_n \to 0\).

The book originates in Italy and is based on many years of written tests at the University of Genova. The book’s original goal was to prepare students to succeed in first-year calculus in Italian universities, but it is now being marketed as a world book. The authors apologize in a few spots for an Italy-centric view, but really these points are very minor and everything here would also work in US curricula.

One big weakness of the book is that it makes no mention of technology. The whole first chapter discusses how to draw graphs without mentioning that there are machines today that can do this for us. In many of these problems (especially on sequences and series) it would be very helpful to work out some samples to help guess a good approach. Schaum does a much better job here, and often asks the student to use a graphing calculator to try things out.

Bottom line: a good alternative to Schaum’s Outline for more ambitious students.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

See the table of contents in the publisher's webpage.