In Calculus of One Variable, Sotiros C. Persidis has placed great emphasis on learning the subject through the art of solving problems and techniques of proofs. The text, while very thin, contains a wealth of information designed to prompt quick recollection of topics being learned in a Calculus class. As this is an Alive Book, the extra online documents serve as a great additional resource for student use to help reinforce concepts. I see the online materials as being a source of drill and practice. Students can use these valuable resources for practice with proofs. More practice proofs can only make one better at proofs, solving problems, and learning what to look for when solving problems.
In the United States I mostly see this text being used by mathematics majors who are taking Functions of a Real Variable their junior or senior year as an undergraduate student and in a graduate level setting. I don’t see the text being used as a primary text in a course but as a great supplement. The text can also be used by the professor who wishes to use problems and proofs for their lesson planning to further critical thinking among students. In addition, the text promotes group study sessions where students can take turns teaching and questioning other student’s material in preparation for exams. For students who are studying for a comprehensive or graduate qualifying exam, I see this book as a valuable resource.
The text is to the point. It is not designed to teach concepts from scratch. The readers of this text should have knowledge of Calculus of One Variable before reading this text or at least be learning Calculus of One Variable in parallel with this text. It is a not a text designed for a lastminute cramming session before an exam, in short, not a “Cliff’s Notes” book. However, the text promotes some good study skills for today’s students to note. For example, highlighting is often used in the book to show students what is important.
The use of lists of important properties makes for quick memory recall. For example,

page 34: important limits,

page 36: pictures of discontinuous functions,

page 63: indefinite integrals of elementary functions,

page 64: applications of integration,

pages 70–71: tests for convergence,

pages 80–85: theorems on generalized integrals.
My comments for the text are minimal. On page 3, Fig. 1.1 should be moved to page 2 after the word transcendental. In addition, various numbers should be used and labeled on the real number line. Then, discussions of the absolute value, exponents, and logarithms should follow. I believe this would be a better sequence of discussion.
I would like to see one or two examples of Remarks of Graphs on page 23 being drawn in the text. For example, a graph of (f(x) = (x  2)^2 + 3) or (g(x) = 5left(frac23
ight)^x4) so students can have a quick recall of these properties.
For the discussion of the Library of Functions, perhaps present some of the major functions heavily used. For example, ( f(x) = x^2, x^3, x, sqrt{x},sqrt[3]{x}, ab^x, e^x, log_b(x), ln(x) ) with their graphs and domains and ranges. Connections to vertical and horizontal asymptotes with limits can be made.
I would like to see additional topics in Numerical Analysis added to the text, such as Simpson’s and Trapezoidal Rules, finding areas between two curves using definite integrals, and applications to Ordinary Differential Equations. I believe this would give the book a more diverse appeal to engineering students and further interest to numerical analysis students.
Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362^{nd} Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.