# Everyday Calculus: Discovering the hidden math all around us

As a mathematician, it will not be difficult to connect mathematics to everything you see around you. Look at a tree and you may think about complex fluid dynamical modeling problems for the lifestream pumped from the roots to the leaves. If you look out of the window, you may associate it with the control problem at the factory to produce flawless glass panes.

Fernandez does something similar, although at a much less advanced level of mathematical complication. In fact he follows the backbone of an elementary calculus course: functions, limits, continuity, derivatives, curvature, (simple) differential equations (mathematical models), optimization, integrals and their applications. As he walks through the calculus course, he describes in parallel an ordinary day in his life and with every event or moment he can associate a mathematical issue. The level of the mathematics is elementary, although sometimes he lifts it to a somewhat higher level. So I am not sure what kind of public will appreciate this book most. If as a reader I am a true mathematical virgin, then it is nice to see how changes in stock markets can be used to evenually lead to the definition of a derivative, but if I am interested in stock markets, I will probably not be a mathematical virgin. If on the other hand I do know my calculus course, then I would probably be a bit bored by the time it takes to finally arrive at the derivative which I already know. So I guess that the readership that will appreciate this most will probably be the people who did have a calculus course long ago, but who have forgotten most of it, and never understood why they had to spend so many hours of their youth studying mathematics, that they never needed or made use of afterwards.

It takes not a lot of imagination to link almost any application you have in mind to almost any instant of and average day of your life. If there is nothing triggering the application in a visual of auditive way, it suffices to say that looking as X, it makes you think of Y, and there you are. So the fact that Fernandez is describing an average day in his life is not very important, it's just a backbone on which to hinge the successive chapters, which happen to be the successive chapters in a calculus course. The originality is just the applications chosen to illustrate the mathematics.

Let me take the first chapter as an example. When waking up, a discussion is given about the periodic REM sleep as a sinusoidal phenomenon. The alarm can be an alarm clock working on AC current or an iPhone working on DC current. That triggers a rather elaborative discussion on electricity and why AC has defeated DC when it comes to transporting electricity over longer distances. Think further of radio waves and audio waves decaying at a logarithmic rate, and think of the parabolic path of the jet of water from the shower under influence of gravitation and clearly there you have enough examples to introduce the elementary functions: trigonometric, logarithmic, polynomials,... The other chapters are similar, gradually increasing the mathematical level. Chapter two is breakfast, listening to increase/decrease on stock market products is about changes, i.e., derivatives. Similarly the cooling down of the coffee, or the deterioration of the effect of vitamin pills is about the derivative shaping the decay. Other (traditional) examples are the speed of a raindrop under gravity with air resistance, the change of unemployment rate,... An example that one would not expect in an elementary calculus course is the correction needed for a GPS signal because of Einstein's relativity theory. Since the GPS signal travels at the speed of light, relativity requires a correction to calculate the spatial position, otherwise it would introduce an error of about 1.34 miles per day (all quantities are expressed in American units, no standard metric system in this book). Some other topics discussed: spam mails reducing productivity, the chance of catching a cold from somebody, the sustainable amount of fish to catch, the optimal angle at which our blood vessels are branching, the optimal pricing of a theater ticket, traffic speed control over a road section. The introduction of Riemann sums and the integral is kept rather mathematical until it comes to the applications: a thermostat, an optimal seat in the movie theater, the length of a train track, the expanding universe. A nice feature of having different applications is that Fernandez shows that the same mathematics and even the same formulas will provide answers to problems that seem to be totally unrelated at first sight.

Some details about steps made in the main text are further elaborated in appendices. There is a confusing typo on page 55 where it says that some graph is curving upward (convex) and later curving downward (concave). This curvature is expressed by the second derivative (first positive, later negative), but it is given the wrong sign in the book.

To conclude we can say that the author is trying very hard to show that mathematics is all around us. In that respect he certainly succeeded. It's a laudable achievement that he brings this with illustrations that are both modern, nontrivial, even including relativity theory and cosmology. As a mathematician he is emphasizing the fun of realizing that the math is really able to model and solve these (simplified) problems. As a mathematician one may easily understand these feelings and enjoy it as much as Fernandez does. However, when reading this with the eyes and mind of a youngster, who feels annoyed and bored with all these mathematics, and who does not care about all the mathematics governing whatever he is doing, which he will do anyway, whether or not he understands the underlying mathematics, I am not sure that such a student will experience the same 'fun' as the author experiences. What I am missing a bit is the playful element that generates the true 'fun' for the not-yet-mathematician, and a bit of the challenge that is needed to really engage the reader. I do not think the book will convince the non-believers, who couldn't care less about all these mathematics. They will not be convinced because the mathematics are too explicit and hence for them the book will be a bit less boring than their lecture notes, but probably still boring. On the other hand, the book is perfect for a reader who really wants to know what mathematics are governing our lives and who wants to learn and understand or polish up his rusty knowledge of these mathematics.

**Submitted by Adhemar Bultheel |

**12 / May / 2014