Dan Sloughter’s *Calculus from Approximation to Theory* offers a refreshing, high-level introduction to the subject. As the title suggests, the author uses approximations (rather than limits) as his motivating lens, and the text highlights this angle by including discourses on numerical techniques and careful study of the order of approximations. This change of motivation prompts a reordering of some topics; however, the text sufficiently covers the fundamental material in a typical single-variable calculus curriculum. The result is a calculus textbook that is at times approachable, at times demanding; and that breaks free from the typical calculus mold. I would recommend this text for use in a calculus course geared towards math majors, for an honors calculus course, or for strong students to use for independent study.

The relationship between approximation, limits, and exactness is at the heart of calculus. Sloughter’s text explicitly identifies this, describing calculus as “a suite of techniques that approximate curved things by flat things”, and going on to discuss how a limiting process can be used on these approximations to “arrive at an exact answer.” This book explicitly “places its emphasis on the approximating process,” as opposed to standard approaches, which “focus on that limiting process as the heart of the matter.” This choice reverberates throughout the text, changing the order in which the material is introduced and reframing how the material is presented.

On the ordering of topics: The most notable departure from the standard calculus framework is the subject of sequences, sequential limits, difference quotients, and (basic) series: these topics are introduced in the first chapter of the text. Sequences motivate the limiting/approximation process and are used over the next several chapters to support the exploration of the usual calculus topics (chapters 2,3, 4, and 5 focus on functions, derivatives, integrals, and Taylor series respectively). Rounding out the contents of the text, we have a chapter on transcendental functions (including the exponential and logarithmic functions, which are reserved until this moment), a chapter introducing the complex plane (which includes a robust discussion of the two-body problem), and a chapter on differential equations. Many of the “usual” topics in calculus find a natural home in this text, although their location might be different than what one is used to: for example, l’Hôspital’s rule is saved until a chapter on limit calculations after power series have been introduced.

On the presentation of material: Sloughter is careful and precise in how he discusses approximating techniques. This leads to a text that doesn’t shy away from technical mathematical language. The “epsilon-N” definition of a sequence is introduced early and used in a fundamental way, as is the language and use of “big-O” and “little-o” notation. Sections that might be reserved for an “optional” section in a standard calculus text are fundamental in this text. The prose of the text is friendly, but is not overtly coddling; it is the kind of prose I associate more commonly with an undergraduate analysis textbook. Each section of each chapter has only a handful of exercises, but these exercises seem to be carefully and well chosen. (compare to the usual Calculus textbooks, which feature dozens of interchangeable computational problems per section).

Overall, I am very impressed with this text. It offers a comprehensive, somewhat unusual approach to the topics of calculus, and leans into this approach with gusto. In many ways, this calculus book is more demanding than many; but students who engage honestly with this will benefit tremendously. I worry that its aim -- more rigorous than many Calculus texts, but not as robust as some Analysis texts – might make it challenging to find a natural place in the undergraduate curriculum. But, for the right student or the appropriate honors calculus class, this text should absolutely be considered.

John Ross is an assistant professor of mathematics at Southwestern University