Calculus: Early Transcendentals

This calculus text is a project of Lyryx.com. The entire text is available as a free PDF download. In addition, there is an online formative evaluation system (similar to WebAssign or WebWorks) as well as a course management system. The cost for the evaluation system is under $40 per student. There is also a free version which requires students to use on campus computers. I tried the sample questions and found the questions to be very well designed. In addition, the feedback provided for wrong answers seemed quite helpful. Each time you activate the system a new set of questions is randomly generated. Instructor support as described on the Lyryx website includes

• Demo questions to use as a teaching tool
• Multiple Choice question bank

• Hard-copy exercises, and solutions to selected exercises
•  Beamer slides (available with Mathematics products): TeX slides aligned to the selected content, which are fully customizable for your classroom.

I encourage you to visit the website for more details. The only other math text Lyryx offers is Linear Algebra.

The text itself presents the standard material for a two to three semester calculus course in a (by modern standards) very brief 577 pages. The chapter headings match up well with Stewart’s 8th Edition which runs over 1100 pages. At North Central, we use a loose page version of Stewart with WebAssign access — the price for students is $168.75. So what is missing? Is it important enough to warrant the substantial difference in price?

Here are a few things I found:

• The number of examples and exercises is substantially less in the Lyryx text. For example, Stewart has 50 exercises on related rates, Lyryx has 16. Stewart has 82 optimization problems, Lyryx has 28.
• There are no projects nor are there many advanced problems in the Lyryx text.
• There are no biographical notes in the Lyryx text.
• The text is entirely in black and white and lacks the high-quality color graphics of a main line calculus text. This is particularly telling in the chapters on functions of several variables and vector functions. Given the availability of online graphing routines this may not be that big an issue. As an example, 3-D Function Plotter provides very nice plots of functions of the form \(z=f(x,y)\) as well as their associated contour plots.

In summary, this is a bare bones text and would probably not be suitable for honors or even very well prepared students. On the other hand, it is clearly pitched at too high a level for a survey calculus course. That being said, it is probably at just the right level for the vast majority of students that many of us deal with. For students majoring in mathematics, a bridge course could fill in the gaps while providing an introduction to proof techniques.

The text begins with a two-chapter review of precalculus material. These chapters provide a very nice summary of the important information needed to succeed in calculus. Along with the online evaluation materials (including labs and quizzes), this should allow instructors to customize their courses based on student background. In particular, this text would work very well for a combined precalculus/calculus course.

The chapter on limits is well-done and includes the formal definition and one non-trivial \(\varepsilon\)-\(\delta\) proof. Some of the theorems are hard to read due to the formatting. One example is L’Hôpital’s Rule where the entire theorem including conditions and result is on one line.

I found the treatment of extrema to be problematic. In Section 5.2, extrema are located and classified using values of the function near a point at which \(f'(a)=0\) or fails to exist. The first derivative test does not appear until Section 5.6.1 on curve sketching and is not labeled as such. The first example references the First Derivative Test — I suspect students will be confused. The Second Derivative test is described in a fairly long paragraph — again, a likely source of confusion. As mentioned earlier the optimization section has only 28 exercises (versus 82 in Stewart) and no illustrations are provided for the exercises.

The treatment of the limit of sequences and series is missing a few key ideas. In particular, he limit comparison test for infinite series is not presented. In addition the fact that any finite number of initial terms can be ignored when using the comparison test is stated incorrectly.

The stated premise for Theorem 9.38 is “If \(a_n, b_n\) are non-negative for all \(n\) and \(a_n\leq b_n\,\forall n\geq N\).” I think it should be stated as: If \(0\leq a_n\leq b_n\,\forall n \geq N\). In addition, at least one example of ignoring the first few terms should be provided.

The text’s statement of the comparison test makes a problem like 9.5.3, \[\sum_{n=1}^\infty \frac{1}{2n^2-3n-5},\] very difficult to deal with. The first two terms are negative and every term after that is larger than the obvious comparison sequence. We can clearly ignore the first few terms, but the Comparison Theorem (p. 360) would not apply. Finally, there is no discussion of how to extend standard series to functions that do not have closed-form antiderivatives (such as \(e^{x^2}\)). That seems to me to be an important omission.

The chapters on multivariable calculus suffer from a lack of high-quality color graphics. For example, there is a sketch (p. 502) of the intersection of three cylinders that is very hard to interpret. There are also some issues with presentation similar to those in earlier sections. The formula for the directional derivative appears only at the end of its derivation. I think important results like this need to be in a box and labeled. A second example is the second derivative test for extrema (p. 481). This result is presented as a single paragraph making it difficult to read.

Cylindrical and spherical coordinates in 3-space receive a very brief treatment as is the case with the discussion of the Jacobian. Once again, the key formulas are as easy to find as I think they should be. Students (for better or for worse) expect to find key ideas inside boxes — I do not think the advantages (if any) of making students work to find them outweigh the impact of the frustration caused by forcing students to hunt for them.

The discussion of vector-valued functions is clear and concise. I would, however, prefer that the definition of the line integral be written out in more detail at the beginning. Instead of \( \int_a^b F\cdot r'\,dt\) I think we need to make things more explicit: \(\int_a^b F(r(t))\cdot r'(t)\,dt\). The authors do this for the line integral of a scalar function.

Overall, I found this text to be clearly written and (with a few exceptions, some of which are discussed above) easy to read. The addition of boxed in formulas and some more challenging problems would make this a much better text.

I requested a sample course, and it appeared within an hour! The online system seems easy to navigate and the lab/quiz questions seem to be well constructed and are aligned well with the course content. I think the text, along with its accompanying online course materials could be a good choice for a standard calculus course designed for all but the best students. There are not as many problems as a standard text, but I suspect that many faculty do not assign more problems than are presented in the Lyryx text anyway. Instructors who want to assign projects and problems that are more difficult would need to find them elsewhere. I would encourage faculty considering a textbook change to investigate this option. I would also recommend this text to anyone wishing to learn (or review) calculus on their own.

Richard Wilders is Professor of Mathematics, and Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences at North Central College in Naperville, IL. He has taught calculus for over 40 years and still finds it exciting! rjwilders@noctrl.edu.