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
This post is part of a series, explore additional posts hereWhat is Calculus?: Limits >>