# Calculus Simplified

Oscar Fernandez is the author of Everyday Calculus: Discovering the Hidden Math All around Us (2014) and The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health (2017) which are extracurricular texts to illustrate the applications and usefulness of calculus and mathematics in general. With the current book he provides the actual lecture notes, or, as he defines it: a calculus supplement. He starts the book with an extensive commercial, as if he has to justify that he is adding "yet another calculus supplement" to what is already available. His main argument is that he uses the "goldilocks approach", i.e., he provides everything in "just the right amount": just the right level of abstraction, details, insight, intuition, applications, number of pages,... Fernandez characterizes his book as a goldilocks average of a proper calculus text, a calculus supplement, and a calculus teacher. Thus the book provides all the goodies of the usual calculus texts in combination with the help that a teacher would add to it: highlighted text, frames, summaries, take home messages, tips and tricks, many exercises and solutions, and a PUP website for interactive content.

All this information is what you usually expect at a publisher's website or on a flyer advertising the book, so it is a bit strange to read it in a book that you already have purchased, but it has the advantage that you are well informed about what and what not to expect, even before you start reading. In fact, "reading" is not the right verb as Fernandez correctly advises the reader to "work through" the book rather than just read it. The level is elementary, somewhere between pre-calculus and first year calculus. The difference between both approaches is static versus dynamic as Fernandez explains: for example pre-calculus just gives the formula for the volume of a sphere (static), while calculus explains the formula as the limiting value as a sum of discs that become infinitesimally thin (dynamic). The infinitesimal concept (something becoming infinitely small without ever being zero) is essential for practical concepts such as instantaneous speed, slope of a curve, and area of a region, which relate to the calculus concepts of limit (the foundation of it all), continuity, derivative, and integral. These are the traditional mathematical topics to be expected but Fernandez manages to cover this in only 109 pages (excluding exercises). The exponential, logarithmic and trigonometric functions are optional, so that everything can be treated using only algebraic functions. At many places a section entitled Transcendental Tales is inserted where the general theory is applied to these transcendental functions. If you skip these sections, then only 89 pages suffice. In fact the main text ends after 158 pages (that includes the exercises). This means that about one third of the main text consists of exercises. The rest of the book is mainly a set of appendices surveying all the material that is supposed to be known in advance (algebra, geometry, functions), answers to exercises, additional applications, bibliography, and index, all together these add an extra 90 pages.

Elementary as the subjects may be, what has been treated has all the rigour that one would expect. There are definitions and there are theorems, but proofs are skipped or hidden in an appendix or in an exercise or it is replaced by several illustrating examples. Examples and applications are main ingredients of the text. Especially optimization as an application of differentiation gets a separate chapter and is rather well elaborated.

I do not think this is suitable for mathematics or engineering students. These definitely need more depth. Unless they are at a pre-calculus level and are so eager that they want to learn more calculus in advance on their own. However, for students that will need some mathematics, and are required to take a calculus course, even if they do not like it, then this book is a nice approach, indeed for the reasons given by the author in his introduction. Many glossy calculus books of up to a thousand pages are a major overkill. The many examples here should stimulate intuition before rigour. Not including the proofs is a practice that has gained popularity, probably not to the liking of mathematicians, but it might help students that are abhorred by the required formal and technical details of a proof. The PUP website is not spectacular but it works nicely and smoothly. There the "dynamic" approach of calculus is lively illustrated by the animations. The text can be "personalized" by skipping for example the sections Transcendental Tales and/or some applications. It will require the guidance of a teacher to help make the proper decisions. Making a selection is however something that one can do with every text. The most interesting property of this book is in my opinion the conciseness (which need to be taken with a grain of salt, since it depends on how much one is prepared to skip, and how many of the exercises are considered to be essential). The abundance of examples instead of proofs is another distinct property, but I believe that also exists in some other texts (or one could just skip the proofs if they are present). Thus I believe there is indeed a wide potential readership for this text since the chosen ones among the students that love mathematics with all its rigour, proofs and technicalities is still a minority with respect to all the students that are submitted to a calculus course because they just need a minimal amount of mathematics.

**Submitted by Adhemar Bultheel |

**1 / Jul / 2019