A Gentle Introduction To Learning Calculus (2024)

I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education.

Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). It all fits together.

Calculus is similarly enlightening. Don’t these formulas seem related in some way?

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They are. But most of us learn these formulas independently. Calculus lets us start with “circumference = 2 * pi * r” and figure out the others — the Greeks would have appreciated this.

Unfortunately, calculus can epitomize what’s wrong with math education. Most lessons feature contrived examples, arcane proofs, and memorization that body slam our intuition & enthusiasm.

It really shouldn’t be this way.

Math, art, and ideas

I’ve learned something from school:Math isn’t the hard part of math; motivation is.Specifically, staying encouraged despite

  • Teachers focused more on publishing/perishing than teaching
  • Self-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject”
  • Textbooks and curriculums more concerned withprofitsand test results than insight

“…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”

Imagine teaching art like this:Kids, no fingerpainting in kindergarten.Instead, let’s study paint chemistry, the physics of light, and the anatomy of the eye. After 12 years of this, if the kids (now teenagers) don’t hate art already, they may begin to start coloring on their own. After all, they have the “rigorous, testable” fundamentals to start appreciating art. Right?

Poetry is similar. Imagine studying this quote (formula):

“This above all else: to thine own self be true, and it must follow, as night follows day, thou canst not then be false to any man.” —William Shakespeare, Hamlet

It’s an elegant way of saying “be yourself” (and if that means writing irreverently about math, so be it). But if this were math class, we’d be counting the syllables, analyzing the iambic pentameter, and mapping out the subject, verb and object.

Math and poetry are fingers pointing at the moon. Don’t confuse the finger for the moon.Formulas are ameans to an end, a way to express a mathematical truth.

We’ve forgotten that math is about ideas, not robotically manipulating the formulas that express them.

Ok bub, what’s your great idea?

Feisty, are we? Well, here’s what I won’t do: recreate the existing textbooks. If you need answersright awayfor that big test, there’s plenty ofwebsites,class videosand20-minute sprintsto help you out.

Instead, let’s share the core insights of calculus. Equations aren’t enough — I want the “aha!” moments that make everything click.

Formal mathematical language is one just one way to communicate. Diagrams, animations, and just plain talkin’ can often provide more insight than a page full of proofs.

But calculus is hard!

I think anyone can appreciate the core ideas of calculus. We don’t need to be writers to enjoy Shakespeare.

It’s within your reach if you know algebra and have a general interest in math. Not long ago, reading and writing were the work of trained scribes. Yet today that can be handled by a 10-year old. Why?

Because we expect it. Expectations play a huge part in what’s possible. Soexpectthat calculus is just another subject. Some people get into the nitty-gritty (the writers/mathematicians). But the rest of us can still admire what’s happening, and expand our brain along the way.

It’s about how far you want to go. I’d love for everyone to understand the core concepts of calculus and say “whoa”.

So what’s calculus about?

Somedefine calculusas “the branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables”. It’s correct, but not helpful for beginners.

Here’s my take: Calculus does to algebra what algebra did to arithmetic.

Using calculus, we can ask all sorts of questions:

  • How does an equation grow and shrink? Accumulate over time?
  • When does it reach its highest/lowest point?
  • How do we use variables that are constantly changing? (Heat, motion, populations, …).
  • And much, much more!

Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them.Like evolution, calculus expands your understanding of how Nature works.

An Example, Please

Let’s walk the walk. Suppose we know the equation for circumference (2 * pi * r) and want to find area. What to do?

Realize that a filled-in disc is like a set of Russian dolls.

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Here are two ways to draw a disc:

  • Make a circle and fill it in
  • Draw a bunch of rings with a thick marker

The amount of “space” (area) should be the same in each case, right? And how much space does a ring use?

Well, the very largest ring has radius “r” and a circumference 2 * pi * r. As the rings get smaller their circumference shrinks, but it keeps the pattern of 2 * pi * current radius. The final ring is more like a pinpoint, with no circumference at all.

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Now here’s where things get funky.Let’s unroll those rings and line them up.What happens?

  • We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles).
  • One side has the smallest ring (0) and the other side has the largest ring (2 * pi * r)
  • We have rings going from radius 0 to up to “r”. For each possible radius (0 to r), we just place the unrolled ring at that location.
  • The total area of the “ring triangle” = 1/2 base * height = 1/2 * r * (2 * pi * r) = pi * r^2, which is the formula for area!

Yowza! The combined area of the rings = the area of the triangle = area of circle!

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This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings.

This is a recurring theme in calculus:Big things are made from little things.And sometimes the little things are easier to work with.

A note on examples

Many calculus examples are based on physics. That’s great, but it can be hard to relate: honestly, how often do you knowthe equation for velocityfor an object? Less than once a week, if that.

I prefer starting with physical, visual examples because it’s how our minds work. That ring/circle thing we made? You could build it out of several pipe cleaners, separate them, and straighten them into a crude triangle to see if the math really works. That’s just not happening with your velocity equation.

A note on rigor (for the math geeks)

I can feel the math pedants firing up their keyboards. Just a few words on “rigor”.

Did you know we don’t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because“Sure, it works in practice. But does it work in theory?”.

We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process.

We’re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature’s way of saying “This has lots of energy. Eat it.”

I don’t want to (and can’t) teach an analysis course or train researchers. Would it be so bad if everyone understood calculus to the “non-rigorous” level that Newton did? That it changed how they saw the world, as it did for him?

A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but theintuition of growthis how it was discovered. The natural log can be seen as an integral, or thetime needed to grow. Which explanations help beginners more?

Let’s fingerpaint a bit, and get into the chemistry along the way. Happy math.

(PS: A kind reader has created ananimated powerpoint slideshowthat helps present this idea more visually (best viewed in PowerPoint, due to the animations). Thanks!)

Note: I’ve made an entire intuition-first calculus series in the style of this article:

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A Gentle Introduction To Learning Calculus (2024)


A Gentle Introduction To Learning Calculus? ›

In mathematics, basic calculus is the mathematical field of study concerning continuous change as well as how things change. In practice, basic calculus refers to the study of functions and limits and is broken down into the two major branches: Differential calculus – This deals with derivatives and differentials.

How can a beginner learn calculus? ›

Best Way to Learn Calculus!
  1. Step 1 Begin with Other Basic Parts of Mathematics. ...
  2. Step 2 Know the Parts of Calculus. ...
  3. Step 3 Learn Calculus Formulae. ...
  4. Step 4 Know the Concept of Limits. ...
  5. Step 5 Understand the Fundamental Theorem of Calculus. ...
  6. Step 6 Practice More and More Calculus Problems. ...
  7. Step 7 Ask your Doubts.

What is the basic calculus introduction? ›

In mathematics, basic calculus is the mathematical field of study concerning continuous change as well as how things change. In practice, basic calculus refers to the study of functions and limits and is broken down into the two major branches: Differential calculus – This deals with derivatives and differentials.

What is the first thing you learn in calculus? ›

Limits are a fundamental part of calculus and are among the first things that students learn about in a calculus class. In short, finding the limit of a function means determining what value the function approaches as it gets closer and closer to a certain point.

How can I make calculus easier to understand? ›

8 Tips on How to Study for Calculus
  1. Solve Calculus Problems Strategically. ...
  2. Take Excellent Notes. ...
  3. Go Back to Precalculus Lessons. ...
  4. Include as Many Drawings in Your Notes as Possible. ...
  5. Seek External Visualizations. ...
  6. Keep a Dictionary of Calculus Notations and Terms. ...
  7. Think of Real-World Applications for Calculus Tools.
Nov 2, 2022

Can the average person learn calculus? ›

Yes, it will take hard work at times, but the numerous benefits you'll obtain when you master it are unrivaled. It will reveal things to you that are hidden from most people's eyes. Believe in yourself, because anybody can “do calculus.” So, take a deep breath, get started, and be ready to expand your mind.

How long does it take to self teach calculus? ›

The learning duration varies based on proficiency levels and individual factors. Basic proficiency may take six months to a year, intermediate proficiency about two years, and advanced proficiency several years.

What is the first rule of calculus? ›

The first fundamental theorem says that the value of any function is the rate of change (the derivative) of its integral from a fixed starting point up to any chosen end point.

What are the 3 main concepts of calculus? ›

The main concepts of calculus are : Limits. Differential calculus (Differentiation). Integral calculus (Integration).

What is calculus in layman's terms? ›

In simplest terms, calculus is a branch of mathematics that deals with rates of change. For example: maybe you want to calculate the change in velocity of a car rolling to a stop at a red light. Calculus can help you figure out that change.

What should I know before I take calculus? ›

Know how to manipulate polynomial expressions. Know how to solve simple linear equations. Know the properties of exponents. Know what logarithms are, as well as their properties.

Is calculus easier than algebra? ›

Calculus is the hardest mathematics subject and only a small percentage of students reach Calculus in high school or anywhere else. Linear algebra is a part of abstract algebra in vector space. However, it is more concrete with matrices, hence less abstract and easier to understand.

What is the hardest type of math? ›

Differential equations, real analysis, and complex analysis are some of the most challenging mathematics courses that are offered at the high school level. These courses are typically taken by students who are interested in pursuing careers in mathematics, physics, or engineering.

Why do so many students struggle with calculus? ›

Calculus is widely regarded as a very hard math class, and with good reason. The concepts take you far beyond the comfortable realms of algebra and geometry that you've explored in previous courses. Calculus asks you to think in ways that are more abstract, requiring more imagination.

What's so hard about calculus? ›

Students also find this kind of math to be difficult because of the unfamiliarity of the concepts they are aiming to calculate in their work. In calculus, students will be asked to examine rates of change by introducing concepts like limits, derivatives and integrals.

Is calculus hard for beginners? ›

Despite being a fundamental subject in the field of mathematics, calculus is notorious for its difficulty. Many students struggle to learn calculus and find it to be a daunting subject.

Which calculus should I learn first? ›

Precalculus. The standard prerequisite for freshman-level calculus is three years of high school mathematics, including trigonometry and logarithms. Students who need to take calculus but are lacking the necessary prerequisites should start with a precalculus course.

What age should you start calculus? ›

Researchers in cognitive science assert that by this age, the brain has attained a level of maturity that enables abstract and complex reasoning. The readiness of the adolescent mind, coupled with a foundation in basic mathematical principles, makes the age of 14 an opportune time to introduce calculus.

How can I be good at basic calculus? ›

You need to practice solving problems in the right way. Try to solve problems in front of the professor, TA, tutor or even peers. This way, you can get immediate feedback and the practice is much more likely to be “good” practice. You cannot master calculus by looking at other people's work—or the solutions manual.

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