What is Calculus?

By Oscar Fernandez

This is the first of three short articles exploring calculus. This article briefly explores its origins. The second and third articles explore its substance and impact, respectively. They will be published in the coming weeks.

What is calculus? If you were watching Jeopardy on May 31, 2019 you were treated to one whimsical answer: “developed by 2 17th century thinkers & rivals, it’s used to calculate rates of change & to torment high school students.” Funny, Jeopardy. While that answer isn’t totally accurate, what I do like about it is its structure—history, substance, and impact. This is a tried-and-true powerful framework for understanding new concepts that marries context with content. In this three-part series on calculus I’ll give you a short introduction to calculus’ history, substance, and impact to provide you with a more fulfilling answer to the question “what is calculus?” First up: a short tour of the origins of calculus.

Three Big Problems That Drove the Development of Calculus

By the mid-1600s, scientists and mathematicians had spent millennia trying to solve what I’ll call the three Big Problems in mathematics: the instantaneous speed problem, the tangent line problem, and the area problem. The figure below illustrates these.

(Reprinted, with permission, from Calculus Simplified (Princeton University Press))

The instantaneous speed problem (a) popped up in many places, most notably in connection with Isaac Newton’s studies of gravity. You see, gravity continuously accelerates a falling object, changing its velocity from instant to instant. To fully understand gravity, then, requires an understanding of instantaneous velocity. This didn’t exist before calculus. The tangent line problem (b) arose mainly as a mathematical curiosity. The ancient Greeks knew how to calculate tangent lines to circles, but until calculus no one knew how to do that for other curves. The area problem (c) popped up in a variety of places. Ancient Egyptian tax collectors, for example, needed to know how to calculate the area of irregular shapes to accurately tax landowners. Many hundreds of years later, the ancient Greeks found formulas for the areas of certain shapes (e.g., circles) but no one knew how to find the area of any shape.

From understanding gravity to calculating taxes to mathematical curiosities, the three Big Problems illustrate the broad origins of calculus. And for millennia they remained unsolved. What made them so hard was that they could not be solved with pre-calculus mathematics. For example, you’ve been taught that you need two points to calculate the slope of a line. But in the tangent line problem you’re only given one point (point P in (b)). How can one possibly calculate the slope of a line with just one point?! Similarly, we think of speed as “change in distance divided by change in time” (as in “the car zoomed by at 80 miles per hour”). That’s a problem for the instantaneous speed problem (a), because there’s zero change in time during an instant, making the denominator of “change in distance divided by change in time” zero. We can’t divide by zero, so again we’re stuck.

The Two Geniuses That Figured Everything Out

It wasn’t until the mid-1600s that real progress on solving the three Big Problems was made. One thing the Jeopardy answer above got right was the allusion to the two 17th century thinkers credited with making the most progress: Isaac Newton and Gottfried Leibniz. You probably know a few things about Newton—you may have heard about Newton’s Three Laws of Motion, which forms the foundation of much of physics—but you’ve likely heard little if at all about Leibniz. That’s because, in short, Newton used the eventual power and influence he gained after making his many discoveries and advances public to discredit Leibniz’s role in the development of calculus. (Read more about the feud here.) Yet each of these great thinkers made important contributions to calculus. Their frameworks and approaches were very different, yet each provides tremendous insight into the mathematical foundations of calculus and how calculus works.

In the next post in this series we’ll dive into those foundations. We will discuss the ultimate foundation of calculus—limits—and the two pillars erected on that foundation—derivatives and integrals—that altogether constitute the mansion of calculus. And we will discover an amazing fact: all three of the Big Problems can be solved using THE SAME approach. As is true with so many thorny problems, we will see that all that was required was a change in perspective.


Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

Pi: A Window into the World of Mathematics

Mathematicians have always been fascinated by Pi, the famous never-ending never-repeating decimal that rounds to 3.14. But why? What makes Pi such an interesting number? Every mathematician has their own answer to that question. For me, Pi’s allure is that it illustrates perfectly the arc of mathematics. Let me explain what I mean by taking you on a short mathematical adventure.

Picture yourself in a kitchen, rummaging the pantry for two cans of food. Let’s say you’ve found two that have circular bases of different diameters d1 and d2. Associated with each circle is a circumference value, the distance you’d measure if you walked all the way around the circle.

Were you to perfectly measure each circle’s circumference and diameter you would discover an intriguing relationship:

In other words, the ratio of each circle’s circumference to its diameter doesn’t change, even though one circle is bigger than the other. (This circumference-to-diameter number is  (“Pi”), the familiar 3.14-ish number.) This is the first stop along the arc of mathematics: the discovery of a relationship between two quantities.

Where this story gets very interesting is when, after grabbing even more cans and measuring the ratio of their circumferences to their diameters—you seem to have lots of free time on your hands—you keep finding the same ratio. Every. Time. This is the second stop along the arc of mathematics: the discovery of a pattern. Shortly after that, you begin to wonder: does every circle, no matter its size, have the same circumference-to-diameter ratio? You have reached the third stop along the arc of mathematics: conjecture. (Let’s call our circumference-to-diameter conjecture The Circle Conjecture.)

At first you consider proving The Circle Conjecture by measuring the ratio C/d for every circle. But you soon realize that this is impossible. And that’s the moment when you start truly thinking like a mathematician and begin to wonder: Can I prove The Circle Conjecture true using mathematics? You have now reached the most important stop along the arc of mathematics: the search for universal truth.

One of the first thinkers to make progress on The Circle Conjecture was the Greek mathematician Euclid of Alexandria. Euclid published a mammoth 13-book treatise text called Elements circa 300 BC in which he, among other accomplishments, derived all the geometry you learned in high school from just five postulates. One of Euclid’s results was that the ratio of a circle’s area A to the square of its diameter d2 is the same for all circles:

This is close to what we are trying to prove in The Circle Conjecture, but not the same. It would take another giant of mathematics—the Greek mathematician Archimedes of Syracuse—to move us onto what is often the last stop on the arc of mathematics: thinking outside the box.

Archimedes went back to Euclid’s five postulates, all but one of which dealt with lines, and extended some of Euclid’s postulates to handle curves. With these new postulates Archimedes was able to prove in his treatise Measurement of a Circle (circa 250 BC) that the area, circumference, and radius r of a circle are related by the equation:

(You may recognize this as the area of a triangle with base C and height r. Indeed, Archimedes’ proof of the formula effectively “unrolls” a circle to produce a triangle and then calculates its area.) Combining Archimedes’ formula with Euclid’s result, and using the fact that r = d/2, yields:

Et Voilà! The Circle Conjecture is proved! (To read more about the mathematical details involved in proving The Circle Conjecture, I recommend this excellent article.)

This little Pi adventure illustrated the core arc of mathematics: discovery of a relationship between to quantities; discovery of a more general pattern; statement of a conjecture; search for a proof of that conjecture; and thinking outside the box to help generate a proof. Let me end our mathematical adventure by encouraging you to embark on your own. Find things you experience in your life that are quantifiable and seem to be related (e.g., how much sleep you get and how awake you feel) and follow the stops along the arc of mathematics. You may soon afterward discover another universal truth: anyone can do mathematics! All it takes is curiosity, persistence, and creative thinking. Happy Pi Day!


Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Calculus Simplified, Everyday Calculus, and The Calculus of Happiness (all Princeton).