Calculus & Rates of Change

Watch

• The essence of calculus ↗3Blue1Brown
• What's so special about Euler's number e? | Chapter 5, Essence of calculus ↗3Blue1Brown
• AP Calculus Section 2.1 Rates of Change and Limits ↗Jeffrey Smith

Read

• Calculus Made Easy ↗An enduring primer, this volume demystifies the fundamental concepts of calculus with an emphasis on practical application, making the formidable accessible.Silvanus P. Thompson
• Calculus: An Intuitive and Physical Approach ↗Beyond mere computation, this text grounds calculus in its historical and physical contexts, illuminating the 'why' behind its powerful methods.Morris Kline
• What Is Mathematics? An Elementary Approach to Ideas and Methods ↗While broad in scope, its lucid exposition of calculus offers profound insights into the subject's foundational role within the wider mathematical landscape.Richard Courant and Herbert Robbins
• The First Rule of Calculus: We Don't Talk About CalculusAn engaging essay that strips away the mystique of calculus, revealing its simple, powerful core through relatable examples and clear prose.Steven Strogatz
• Infinite Powers: How Calculus Reveals the Secrets of the Universe ↗A captivating modern narrative that traces calculus from its origins to its pervasive influence, demonstrating its indispensable role in understanding the cosmos and daily life.Steven Strogatz

Voices to follow

• Steven Strogatz ↗His lucid prose demystifies the intricacies of calculus and its profound impact on understanding the universe's dynamic rhythms.Applied Mathematician, Cornell University; Author
• Hannah Fry ↗She masterfully illustrates how the mathematics of change underpins everything from urban planning to human behaviour, making complex ideas relatable.Professor in the Mathematics of Cities, University College London; Science Communicator
• Jordan Ellenberg ↗Through engaging narratives, he reveals how mathematical thinking, including the logic of rates and optimisation, illuminates the hidden structures of the world.Professor of Mathematics, University of Wisconsin-Madison; Author
• Eugenia Cheng ↗She champions the beauty and accessibility of mathematical concepts, often simplifying the foundational ideas that give rise to calculus for a broader audience.Mathematician and Scientist in Residence, School of the Art Institute of Chicago; Author

Glossary

• Average Rate of ChangeThe average rate of change is the overall change in a quantity over a specific interval or period, divided by the length of that interval. It gives you a general idea of how fast something changed between two points. For example, if your height grew from 150 cm to 160 cm over 2 years, your average rate of change in height was 5 cm per year.
• CalculusCalculus is a branch of mathematics that studies how things change and accumulate. It helps us understand motion, growth, and how quantities relate to each other over time or space. For example, calculus can help a scientist predict how fast a rocket will be going at a specific moment, or how much water will fill a tank over an hour.
• DerivativeThe derivative is a fundamental concept in calculus that measures the instantaneous rate of change of a function. It tells you the exact slope of a curve at any single point. For example, if you have a graph showing how far a car has traveled over time, the derivative at any point would tell you the car's exact speed at that specific moment.
• FunctionA function is like a rule or a machine that takes an input number, does something to it, and gives you exactly one output number. It shows how one quantity depends on another. For example, if you have a function that says "double the number and add one," when you input 3, the output is 7 (2*3 + 1).
• Instantaneous Rate of ChangeThe instantaneous rate of change is how fast something is changing at one exact moment in time, rather than over a period. It's like looking at the speed on your speedometer right now. For example, if you're riding a bicycle and check your speed at exactly 3:15 PM, that specific speed is your instantaneous rate of change.
• IntegralAn integral is a concept in calculus used to find the total amount or accumulation of a quantity, often by calculating the area under a curve on a graph. It's essentially the reverse process of finding a derivative. For example, if you know the speed of a car at every moment, an integral can help you find the total distance the car traveled over a certain time.
• LimitA limit is the value that a function "approaches" or gets very, very close to as its input gets closer and closer to a certain number, without necessarily reaching it. It's about what happens near a point. For example, imagine walking towards a wall. The wall is the limit of how far you can go forward. You can get incredibly close, but you can't pass it.
• Rate of ChangeRate of change tells you how quickly one quantity is changing in relation to another quantity. It's about how much something goes up or down over a certain period or distance. For example, if you drive 100 kilometers in 2 hours, your rate of change (speed) is 50 kilometers per hour.
• SlopeSlope is a measure of how steep a line is. It tells you how much a line goes up or down for every step it takes horizontally. For example, a steep hill has a high slope, while a flat road has a slope of zero.
• Tangent LineA tangent line is a straight line that touches a curve at exactly one point, and at that point, it has the same slope or direction as the curve itself. For example, imagine a ball rolling down a curved ramp. If you drew a line that just barely touches the ball at one point, that would be a tangent line, showing the direction the ball is moving at that instant.

Threads 7

Where this connects to other fields, and why it's worth knowing.

• Inflation, Deflation & the Business Cycle Economics

  When the news says 'inflation is falling,' prices are still going up, just less fast than before. People hear 'falling' and think things are getting cheaper, but that's a different thing entirely. It's like a car that's slowing down: it's still moving forward, not backing up.

• Emotion & Emotional Regulation Psychology

  You don't feel how good your life is; you feel whether it's getting better or worse right now. That's why winning the lottery stops feeling amazing after a few months, and why even bad news fades. Your emotions track the change, not the level, so no feeling, good or bad, stays loud for long.

• Film & the Moving Image Arts & Design

  A movie is really just a stack of still photos, 24 per second, flashed so fast your eye blends them into smooth motion. Math has a name for gluing together lots of tiny frozen pieces into a flowing whole: it's called integration, adding up infinitely many slivers. So every second at the cinema, your eye is quietly doing calculus.

• Corporate Finance Business

  A rupee ten years from now is worth less than a rupee today, and the further out you go, the more its value fades. Finance calls this 'discounting,' and it shrinks future money along a smooth curve. It's the exact same math as radioactive decay, how carbon-14 fades over time, just run in reverse. Future money is basically decaying money.

• Climate Change Science

  Think of the CO2 in the sky as water filling a bathtub, where our emissions are the tap. Even if we stop opening the tap wider, the water keeps rising as long as any is still flowing in. Most people expect the level to fall the moment emissions stop growing, but it doesn't, and that mix-up is at the heart of climate confusion.

• Population and Migration Geography

  Even after couples start having fewer than two kids each, a country can keep growing for decades. Why? Because a huge crowd of young people is still coming of age and having babies. It's like a giant ship that keeps gliding forward long after you cut the engine; the momentum in the age structure carries it past the turning point.

• Mechatronics Technology

  System dynamics describes how physical quantities change over time, providing a mathematical basis for feedback design.

  Sources: NPTEL — Mechatronics ↗ · ABET engineering program criteria 2025–2026 ↗