Everyday Calculus: Discovering the Hidden Math All Around Us

by Oscar E. Fernandez

Calculus. For some of us, the word conjures up memories of ten-pound textbooks and visions of tedious abstract equations. And yet, in reality, calculus is fun, accessible, and surrounds us everywhere we go. In Everyday Calculus , Oscar Fernandez shows us how to see the math in our coffee, on the highway, and even in the night sky.

Fernandez uses our everyday experiences to skillfully reveal the hidden calculus behind a typical day's events. He guides us through how math naturally emerges from simple observations--how hot coffee cools down, for example--and in discussions of over fifty familiar events and activities. Fernandez demonstrates that calculus can be used to explore practically any aspect of our lives, including the most effective number of hours to sleep and the fastest route to get to work. He also shows that calculus can be both useful--determining which seat at the theater leads to the best viewing experience, for instance--and fascinating--exploring topics such as time travel and the age of the universe. Throughout, Fernandez presents straightforward concepts, and no prior mathematical knowledge is required. For advanced math fans, the mathematical derivations are included in the appendixes. 

Whether you're new to mathematics or already a curious math enthusiast, Everyday Calculus invites you to spend a day discovering the calculus all around you. The book will convince even die-hard skeptics to view this area of math in a whole new way.

Genres: Science, Mathematics, Nonfiction, Stem, Education

168 pages, Hardcover

First published January 1, 2014

About the author

I am an assistant professor of mathematics at Wellesley College. Math has always been a passion of mine, and now that I've written my first book---Everyday Calculus---I'm looking forward to sharing the wonders of math with more than just my students. Read more about me, my book, and my research at my website: www.surroundedbymath.com. Also, check out my my tweets at @EverydayCalc.

Reviews

[Oscar Fernandez · Rating 5 out of 5]

Rather than review my own book, I thought I'd share here some of my reasons for writing Everyday Calculus and also a link for the first chapter.

Mathematics is beautiful--literally. Find a pretty flower, or a picture of the most handsome person you can think of. Now measure the proportions--for example, the ratio of face length to face width. You’ll likely find the “golden ratio” lurking somewhere (approximately 1.618), or perhaps one of its variants. These numbers also show up in other beautiful objects, including the pyramid of Giza, spiral galaxies, and sunflowers. This power of mathematics to connect seemingly unrelated things together, albeit in a sneaky, hidden way, extends far beyond these examples. We may not realize it, but we’re surrounded by math.

Connections like these were some of my original inspirations for writing Everyday Calculus. There are plenty of books out there that discuss the applications of math, but these applications talk about things many of us will never experience--like space shuttle launches or radiocarbon dating--or care little about (like a ladder falling down, the classic "application" of related rates in calculus). I wanted to write instead about the hidden math behind everyday things and events. This way readers would have an instant connection to the mathematics discussed in the book. I also wanted to make the book as accessible as possible. That's why I decided to move the calculations of the math discussed to the appendices and keep the equations in the body of text to a minimum. This structure would also give readers a preview (or refresher) on basic calculus. This is how Everyday Calculus was born.

You can download the first chapter of the book for free on the Princeton Press site. The corresponding appendix (which has the calculations) isn't included, but you'll still get to experience my approach and also read about connections between calculus and our sleep cycle, electronic devices, and morning shower. You can also listen to a couple of interviews I've done about the book by visiting my website. I should also mention that despite our (the editors and I) best efforts, there are some minor typos in the book. I describe them here. Finally, though the book does discuss a fair amount of calculus, it's not meant to be a substitute for a full calculus course. If you're looking to study calculus formally, consider the many free to low-cost resources I wrote about here.

Whatever you decide to do, I hope that you consider studying more math. As the golden ratio example shows, and as I've tried to convince readers in my book, math really is all around you, and with the right resources and motivation I firmly believe that anyone can understand the subject.

Oscar Fernandez

[Jason · Rating 3 out of 5]

As a secondary math teacher, the question "Why do I have to learn this?" comes up a lot. This book does a good job of presenting how various pre-calculus and calculus concepts show up in 'everyday' life, making strong links between math and various sciences - physics, biology and astronomy. There's a healthy dose of mathematical detail that most strong math students should be able to follow and appreciate. I also like the author's ability to "mathematize" what he sees around him, though sometimes it seems a little forced and awkward.

Although an informative read from a pedagogical perspective, it is somewhat lacking in originality both in its content and writing style. The examples presented are typical of those found in textbooks presented in a more "chatty" style without the numbers and calculations. Using the written word and diagram to illustrate the concept of 'change' - the central concept of calculus - is an outdated media given that we now have computer animation and touch-screen technology to help students understand calculus concepts. It would be interesting to see how much this book would appeal to a person who did not already appreciate the power of calculus.

[Kristin Evans · Rating 4 out of 5]

Raylynn would bully me even more if I wrote a review.

[Joshua · Rating 2 out of 5]

This book is an introduction to the calculus of things that many of us experience everyday. The author takes the reader through a hypothetical 'day-in-the-life' of a math professor, and teaches the fundamental ideas of calculus along the way.

I really wanted to like this book more than I did. I think part of the problem was a mismatch of my expectations. I was hoping for something that took things from everyday experience and demonstrated unexpected connections or conclusions through the underlying mathematics. While there is some of that, it is only weakly connected. I hoped for something more like Ian Stewart or the equivalent to Stephen Jay Gould. But, this book is mainly an introduction to calculus through a series of contrived problems.

One thing that is disappointing is the number of errors in the book. Not typos or incidental errors, but flat-out wrong physical explanations; In the section, "why do we survive rainy days", the author claims that terminal velocity occurs because rain drops coalesce!? After reading that (near the beginning of the book), it is hard not to view the remainder more skeptically and see that there really are a lot of errors.

[Roberto Rigolin F Lopes · Rating 4 out of 5]

The fun here tends to infinity! Calculus formalizes/extends some of humans intuitions about our surroundings (emphasis on 'extends'). Since we operate in an unconscious mode most of the time, calculus is fundamental to understand things overcoming our biased intellect. Prof. Oscar unveils this telling an engaging story about calculus sorting out challenges and events we face daily.

[Fred Kohn · Rating 3 out of 5]

This was a fun book to read but I didn't learn much. Most of the equations were over my head and I didn't have the gumption to dig into it enough to learn. But what the book did do was to whet my interest in calculus once again and make me want to dig up the old calc textbook I picked up at half priced books several years ago. So I guess the book did its job.

[Jayw]

math teacher narrates his day and derives formulas of daily life

[C · Rating 3 out of 5]

This book neither teaches you calculus nor offers any insight into the subject. It does give very good examples of how calculus is used.

[loonchies · Rating 3 out of 5]

เรื่องเล่าอธิบายสิ่งที่เจอได้ในชีวิตประจำวันด้วยความรู้ทางคณิตศาสตร์

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อีก 1 เล่มจากกองดองตั้งแต่ 2010
สมัยตอนที่ยังมีความรู้เรื่องแคลคูลัสมากที่สุดในชีวิต
ได้มาอ่านตอนที่ความรู้เหลือแค่ขนาดบวกลบยังต้องใช้เครื่องคิดเลข
ก็ยังพอถูไถไปได้

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เขียนโฆษณาหนังสือได้ดี
~ อ่านไปเธอจะได้รู้นะว่าทำไมหลอดเลือดถึงได้แตกแขนงทำมุมกว้างขนาดหนึ่ง ๆ, แคลคูลัสเกี่ยวกับการประหยัดน้ำมันหรือหาที่นั่งที่ดีที่สุดในโรงหนังอย่างไร ฯลฯ
อยากอ่านต่อเลย แต่พออ่านไปแล้วรู้สึกว่าต้องใช้ความพยายามอย่างมากเลยแม้จะเป็นเรื่องที่น่าสนใจ จนบางจุดถึงขนาดท้อต้องพักก่อน

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แง่ภาษาอังกฤษ ผู้เขียนใช้คำศัพท์ง่าย อ่านสบาย
หนังสือไม่หนา อ่านแบบ ebook ตัวอักษรไม่ใหญ่มากก็ประมาณ 100 หน้าด้วยซ้ำถ้าไม่รวมภาคผนวกที่อธิบายเพิ่มเติมเกี่ยวกับคณิตศาสตร์ในบทนั้น ๆ แบบลึกขึ้น
แต่แง่ภาษาคณิตศาสตร์นั้นไม่ง่ายเลย (แต่คงง่ายสำหรับเด็กม.ปลายและปี 1)

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ถ้าเคยอ่านเรื่องวิทยาศาสตร์รอบตัว
หนังสือเล่มนี้ก็ยังจับทางแนวนั้น แถมเป็นการอธิบายด้วยวิทยาศาสตร์นั่นแหละ แค่โฟกัสมาทางการมองและเชื่อมโยงจากคณิตศาสตร์มากขึ้น
ขยายลึกลงไปถึงที่มาที่ไปและวิธีคิดที่เป็นเบื้องหลัง ~ ซึ่งก็อธิบายได้ด้วยคณิตศาสตร์นั่นเอง

ผู้เขียนเล่าได้สนุกพอใช้ แม้ว่าหลัง ๆ เราคิดว่ามันน่าเบื่อมาก

แบ่งส่วนธรรมดาให้เข้าใจว่ากำลังพูดถึงเรื่องอะไร สรุปว่าเกี่ยวตรงไหน ใช้คณิตมาอธิบายอะไร
และอีกส่วนที่คณิตข้นหน่อย อธิบายว่าคณิตศาสตร์อธิบายสิ่งนั้น ๆ อย่างไร ส่วนประกอบที่นำมาคำนวณมาจากไหน ไปประกอบในสมการอย่างไร สูตรหยึกหยือนี่บอกอะไรเรา ~ เราสรุปอะไรได้จากสมการนี้บ้าง

โดยรวมสรุปเนื้อหาได้น่าสนใจดี

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แม้ว่าผู้เขียนจะเกริ่นไว้ว่า แม้จะไม่มีความรู้เรื่องแคลคูลัสก็สามารถอ่านหนังสือเล่มนี้เข้าใจและสนุกได้นั้น — ก็ออกจะเป็นเรื่องที่ไม่จริงสักเท่าไร ~~ ไม่ต้องรู้เกี่ยวกับแคลคูลัสก็จริง แต่ควรมีความรู้คณิตศาสตร์บ้าง ถึงจะอ่านแล้วปะติดปะต่อสิ่งที่ผู้เขียนพยายามอธิบายได้ไหลลื่น ถ้าไม่มีเลยอาจต้องใช้เวลาทำความเข้าใจนานหน่อย คือถ้าไม่รู้มาก่อนก็ยังคงพอทำความเข้าใจที่มาที่ไปของการคิดได้ผ่านการอธิบายของผู้เขียน

ไม่งั้นคงต้องทำใจอ่านข้ามไปเลย ไม่ต้องเข้าใจลึกมากว่าข้อสรุปมาได้อย่างไร แต่ถ้าแบบนั้นก็คงน่าเสียดาย

ส่วนตัวคิดว่าถ้าอ่านตั้งแต่สมัยจบมัธยมใหม่ ๆ คงเข้าใจได้เร็ว และจะสนุกกว่านี้
นึกเสียดายที่ดองหนังสือไว้
ตอนนี้เหลือเพียงคอนเซปต์พื้นฐานอันบางเบาเลยใช้เวลาอ่านนานมากเลย

ถ้ามองแบบละเอียด ๆ หน่อย สูตรหยึกยือที่เอามาอธิบายเรื่องต่าง ๆ ก็เป็นสูตรง่าย ๆ เอาตัวแปรมาแทนเลยดูหยึกยือเฉย ๆ ตามสไตล์นักคณิตศาสตร์ที่อยากอธิบายด้วยสัญลักษณ์
แต่บางทีมันชวนน่าเบื่อเพราะขี้เกียจนั่งฟื้นความรู้เพื่อจะเข้าใจอย่างถ่องแท้ หรือนั่งตีความส่วนประกอบในสมการที่ยกมา ก็เลยอ่านผ่าน ๆ ไปเลย

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พออ่านไป...ผู้เขียนไม่ได้พูดถึง”แคลคูลัส”เลย แต่จริง ๆ ก็คือกำลังแตะเรื่องพื้นฐานที่เกี่ยวกับมันนั่นแหละ อธิบายแต่ไม่ได้พูดถึง - Gestsunova ~ เป็นอะไรที่ว้าวดี
แต่พอตรงส่วนที่ลึกขึ้นหรือในบทหลัง ๆ ก็เริ่มรู้สึกว่าคณิตจ๋า ศัพท์เฉพาะโผล่มาประปราย ซึ่งช่วงแรก ๆ ของหนังสือไม่ใช่แบบนั้น

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จริง ๆ บางจุดทำให้รู้สึกว่ากำลังเรียนคอนเซปต์คณิตศาสตร์ผ่านเรื่องในชีวิตประจำวัน มากกว่าจะเข้าใจเบื้องหลังเรื่องในชีวิตประจำวันผ่านคณิตศาสตร์

เหมือนว่าตั้งใจก่อนว่าอยากจะอธิบายคอนเซปต์นึง แล้วจึงค่อยไปหาเรื่องมาใส่ว่าคอนเซปต์นั้นมีในเรื่องอะไรได้บ้างมากกว่า

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แต่สุดท้ายหลังจากรู้สึกสะบักสะบอมกับสารที่ได้รับมา ผู้เขียนก็ปิดจบสรุปได้อย่างสวยงามนะ เหมือนได้คำปลอบใจ ได้ข้อสรุปภาษาเข้าใจง่ายอีกที ขอเพียงมีความอดทน

[William · Rating 3 out of 5]

If you are looking for a book that derives some profound insights to mathematics I would like to point out this book is aimed at people who are not necessarily fluent in mathematics. That being said the first couple of chapters felt a bit too simple and lacked insight that you might in other math/science in your daily life science books. the last 2 chapter seemed to rush to an end and ramp up the use of geometry significantly. I would have liked to have seen those chapters expanded a little more.

[KarenO · Rating 3 out of 5]

I think the book could have been improved if the contents of the appendices were incorporated into the body of the book. As it is, it seemed like a math book with its guts ripped out. I enjoyed it but it could have been much better with more elaboration on how the equations were derived--more Math context.

[Anthony · Rating 4 out of 5]

I enjoyed this book. The section on the logistic equation was excellent and I could put it to use straight away; modelling the growth of mould on bread. It does have a few minor mistakes so best to check the errata.

[Lisa · Rating 4 out of 5]

This is more of a 4.5. I loved it and then it took a lot of brainwork to "get" the last part on integration. It was worth the work! Since I'm using this for Summer Book BINGO, I should note that it scares me that math isn't made enjoyable for everyone.

[Nate Hochmuth · Rating 4 out of 5]

Quick read for those who want to see how math is evident in their everyday days. The author gives short descriptions of how math can explain the various phenomena in a typical day.

[Adam · Rating 2 out of 5]

Doug Gordon's two star review explains. I also thought there were too many unnecessary details: "take the T (our light rail system) downtown"--why not just say train or light rail? or "dinner reservations at an Indian restaurant (Zoraida's favorite)"--is he writing to impress his wife? Fernandez seems to be trying to be specific so his audience can picture what he's talking about; and tries injecting some pathos around the edges, but it just doesn't come off right. OK, enough groaning; here's my regular notes on interesting points:

p 79 If highway speeders were caught using MVT (by averaging the speed between two points), then the roads would be safer. Is it the cost of installing two cameras along a stretch of highway that's stopping the state using this idea?

specific examples of six common graphs
p 7 Voltage for the radius of power lines as the rational function (similar to y = 1/x)
p 16 Electromagnetic values for certain things as the trig function (like y = sinx)
p 38 Velocity of a falling raindrop reaching its terminal velocity as the log function (y = lnx)

p 42 Unemployment thought of as a tangent
p 72 Rough equation for artery angle for max blood flow as (roughly) graph of a catenary (initially pronounced like canteen)~for min gravity pull on power lines
p 95 Common PDF for waiting times is the exponential distribution: e^(-x/m)/m, where m is the average waiting time. For example, if m = 5 min, then one minus the integral from 0 to 5 equals about 37% (from p 144, u = -t/5; du = -1/5 dt). This means that you're 37% likely to be waiting for more than five minutes. And so finding probabilities boils down to integrating PDFs (resulting in CDFs; see Read Jack Lion Heart's answer to What is the difference between a probability density function and a cumulative distribution function? on Quora)
p 108 tell planets from starts by comparing bright disks to pointlike dots [Planets don't twinkle. They are disks, like the sun and moon, and shine steadily. However, this is only apparent when weather conditions are not very good.]--http://www.wikihow.com/Tell-the-Diffe...
nearest star (Proxima Centauri) 25 T mi away ~ 4.2 light years ~ looking 4.2 years in the past
our sun is eight 'light-minutes' away

If the average sleep cycle is 90 min, which could be 80-100 min individually, does SleepChart find your specific phase length? Or does even that very?

Vocab
transitive reasoning - making deductions (A -> B & B -> C; A -> C} mutual induction used by Faraday to step down large AC voltage
sound - pressure wave
free parameter - http://math.stackexchange.com/questio...
gravity - curving of space by matter

Figures
49th key A4 on 88-key piano = 440 Hz (f)
tv remote ~ infrared light ~ about 36 kHz
FM radio station -> f in MHz

Summary (p 116)
1. If all is number, then all is mostly functions.
2. Derivatives describe change; wherever there is change, derivatives.
3. Math usually helps understand something better.
4. Math finds ways to connect things that didn't seem connected.
5. Optimize though it.
6. Integrals [are needed to find quantities, and below].
7. Synergize derivatives and integrals for added insight

[tessa · Rating 3 out of 5]

This book is one that I really had to take time reading because a lot of the concepts were difficult for me to understand. I think that it is fun to realize all of the little ways that math is all around us and enjoyed the book.

[Brett T · Rating 4 out of 5]

Few things will more quickly bring terror to the eyes of many people than the equations of that most arcane and alchemical of mathematical disciplines, calculus. It is the Emperor Palpatine to algebra's Darth Vader, ramping up the insidious pairing of numbers and letters to the level that its letters stand not for mere variables, but for equations filled with other variables!

Most of those who do not flee calculus proclaim utter boredom at its existence and wonder when it will ever be of use in the real world. These people had better never meet Wellesley College assistant math professor Oscar E. Hernandez, whose 2014 book Everyday Calculus will answer their questions in detail.

Using his schedule of a "typical morning," Fernandez shows how everything from the amount of rain falling on his umbrella to the traffic levels during his commute can be approximated, if not precisely explained, by setting the situations up as equations which calculus can solve. After explaining a few key terms like "derivatives" and "differential equations," he is off to the races with his examples, explanations and equations.

Fernandez obviously loves both his discipline of mathematics and the job in which he uses it, teaching. He's amusing and works overtime to tune down the jargon volume and use as many everyday words as possible to describe the process of his reasoning. Equation-phobes will be leery of the book, which includes them at every relevant point and which does not do as well as it could at giving them some context to reduce the brain lock.

With that in mind, though, Everyday Calculus is an energetic and excellent introduction to the ideas of calculus and how complicated some of the simplest real-world phenomena actually are. In places it may be a bit of a slog, but it will also reward the perseverant who make the trip.

Original available here.

[Charles · Rating 5 out of 5]

A series of mathematical explanations for everyday events
As he goes through his day, Fernandez looks at ordinary events and concocts mathematical explanations for what happens. Everything from the cooling of hot beverages in the morning to the rate of the transmission of the common cold to estimating wait times when a train is stalled are examined and explained using calculus. I counted over 30 subtitles in the TOC, so the breadth of topics is considerable.

To his credit, Fernandez pulls no formulaic punches in his descriptions: when a differential or integral is needed he uses it, no matter how complex. This is very refreshing. One thing that I find annoying about math books is when the author exhibits a case of “equation shyness,” an unwillingness to include equations in the text for fear of turning off readers (and reducing royalties). Yet, the topic being explained cannot be fully understood any way other than with the use of one or more equations.
If you are a math teacher looking for some everyday phenomena that are explained using calculus to liven up your course, then this is the book for you. There are many selections; you will have no trouble finding something you can use. If you are looking for a good math book, (with math included), then this one will certainly work for you.

This book was made available for free for review purposes and this review also appears on Amazon

[Doug Gordon · Rating 2 out of 5]

I was really disappointed in this book, which I had hoped would be a review of all that I've forgotten about math and calculus since the days when I had to learn it. Maybe if you were to read it while skipping all the equations you might get something out of it, but when trying to understand the details I was really put off by the number of egregious errors. Didn't anybody proofread this?

First off, on page 47, in explaining how GPS works, he states "my GPS unit communicates ... through signals that travel at the speed of light c, a remarkable 11,176,920 miles per second." In what universe?? That would be really remarkable when in fact light travels at about 186,000 miles per second, which is a fraction of 11 million plus. I have no idea where such a precise but wrong value came from.

Then shortly thereafter, on page 52, he presents this amazing equation: "A(10) = $30/10 = $2". Say what? Divide 30 by 10 and get 2? A few pages later on page 55, he reverses the sign of the second-order derivative (it should be >0 before t* and <0 afterwards, not the opposite as stated).

Frankly, after spotting these glaring errors, I stopped trusting anything that followed and pretty much skimmed through to the end (which didn't take long since this is a very short book).

[Tom Schulte · Rating 3 out of 5]

...Most of the observations herein are prose, with an average of around one equation per page. For more ardent math enthusiasts, there are more rigorous appendixes on calculus fundamentals. Fernandez invites the reader along on this work day and telegraphs an enthusiasm for seeing calculus, with hints of differential equations, presented to him. This excitement will telegraph to the math enthusiast becoming acquainted with calculus through the author’s style, which is both lively and confident.

See my entire review at MAA Reviews.

[Aaron Chu · Rating 3 out of 5]

Oscar's writing style in this book is amusing. The book walks me through a day of Oscar's life and introduces me to various calculus concepts. Although for someone with limited calculus background as I am, as much as the appendixes try to help, some of the equations still appear to be frustratingly mysterious, which discouraged me from finishing the 117-page-book sooner.

[Rafael · Rating 4 out of 5]

I honestry rushed to finish this book, I unsderstood by the title that this was a easy to follow math book, well... maybe not for me. I gave it 4 stars because It made me realize how poor my math skills and knowledge are! I assume that better your math background is, easier for you the reading of this book is.

[Stephen · Rating 4 out of 5]

Fun read. This book provides good motivation to learn calculus. The narrative is friendly and accessible and follows the author through one day's worth of mundane activities while he highlights the underlying calculus based math.

[Mark Bates · Rating 3 out of 5]

OK book a little too much rambling.

[Mills College Library]

515 F3637 2014

[Darko Doko · Rating 5 out of 5]

Wonderfull, I m finaly catching up with my lack of math knowledge :-) better late than never haha

[Gary Jasdzewski · Rating 2 out of 5]

I was looking for a primer to get back into calculus, but this was more of a "look at where calculus shows up in the world" than what I was looking for.