What is Calculus?: The Two Pillars

By Oscar Fernandez

This is the third in a series of short articles exploring calculus. The first article explored the origins of calculus, including the three “big problems” that drove calculus’ development. The second article explored limits, the foundation of calculus. This article discusses how limits help us solve the three “big problems” and introduces two of calculus’ pillars: derivatives and integrals.

In the first article in this series I discussed three Big Problems that drove the development of calculus: the instantaneous speed problem, the tangent line problem, and the area problem. I illustrated these via the figure below.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These problems stumped mathematicians for millennia. (We briefly talked about why in the first article.) But their inability to solve these problems—echoing Morpheus in the movie The Matrix—was not due to the techniques they were using; it was due to their mindset.

How a Dynamics Mindset Solves the Three Big Problems

If you’ve read the second article in this series, you’ll remember my first characterization of calculus: calculus is a dynamics mindset. Yet nothing about the figure above says “dynamics.” Every image is a static snapshot of something (e.g., an area). So let’s calculus the figure. (Yep, I’m encouraging you to think of calculus as a verb.)

The figure below takes each Big Problem from the figure above and adds in the dynamics.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These images show apples falling, gray lines approaching a blue tangent line, and areas being swept out. Lots of movement (dynamics)! Moreover, notice that as the central change in each row of the figure gets closer to zero —the quantity ∆t in the first row and ∆x in the second and third rows—the resulting diagram approaches the respective diagram in the first figure in this article. We’ve met this “as ∆t  approaches zero” language before—it’s the language of limits we discussed in the second article! Adding this new revelation to the figure above produces…


Reprinted, with permission, from Calculus Simplified (Princeton University Press. Click to expand.

Finally, expressing our result in terms of equations involving limits yields the final piece of the puzzle…

Notice how each row employs a dynamics mindset to recast the Big Problem (contained in the “limiting picture” column) as the limit of a sequence of similar quantities (e.g., speeds) involving finite changes, changes which pre-calculus mathematics can handle. Specifically:

  • Row #1: The instantaneous speed of the falling apple is realized as the limit of its average speeds  ∆d / ∆t (ratios of changes in distance to changes in time) as ∆t —> 0.
  • Row #2: The slope of the tangent line is realized as the limit of the secant line slopes ∆y / ∆x (the gray lines in the figure) as ∆x —> 0.
  • Row #3: The area under the curve is realized as the limit as ∆x —> 0 of the area swept out from x = a up to ∆x  past b.

Introducing…Derivatives and Integrals

The limit obtained in the second row of the last figure is called the derivative of f(x) at x = a, the x-value of point P. The limit obtained in the third row of the Figure is called the definite integral of f(x) between x = a and x = b. Derivatives and integrals round out the three most important concepts in calculus (limits are the third).

You now have a working understanding of what derivatives and definite integrals are, what they measure, and how they arise from the application of a dynamics mindset to pre-calculus mathematics. The next post in this series will explore the derivative in greater details. We’ll discover that it has a nice geometric interpretation and a powerful real-world interpretation. (The last figure above hints to what these are.) Near the end of this series we will return to these interpretations to illustrate the power of derivatives, using them to help us understand phenomena as diverse as the fate of the Universe and, more pragmatically, how to find the best seat in a movie theater. Stay tuned!


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 here<< What is Calculus?: Limits