Is calculus that important?  NOTES

By Nethara Rajapakse

What is Calculus?

• Calculus is the study of how things change. It’s a powerful branch of mathematics that helps us model the world in motion.

• There are two main types:

• Differential Calculus: Focuses on rates of change. For example, how fast a car is going right now.

• Integral Calculus: Focuses on accumulation. Like how far the car has travelled over time.

• Fun fact: “Calculus” comes from the Latin word for “small stone”—the kind used for counting.

Real-World Applications of Calculus

1. Technology & Algorithms 

1. Google Maps uses derivatives to calculate your speed and suggest the fastest route. Integration helps model traffic patterns across cities.

2. Spotify and Netflix use calculus behind the scenes in machine learning. Their algorithms model changes in your preferences over time to personalise recommendations.

2. Medicine

1. Drug dosage modelling: Calculus helps doctors know how much of a drug is in your bloodstream at any moment, based on when you took it and how your body processes it.

2. Medical imaging: CT scans and MRIs use algorithms powered by integral calculus to build clear images of your body from lots of tiny data points.

3. Climate & Environment

1. 1. 1. Sea level predictions: Scientists use derivatives to track the rate of sea level rise and integration to model the total rise over decades.

      2. Carbon emissions: Integration calculates total CO₂ output from daily emissions, helping countries plan for carbon neutrality.

4. Engineering & Architecture

1. 1. 1. Bridge and building design: Engineers use calculus to calculate load, stress, and material efficiency.

      2. Roller coasters: Calculus helps model the shape of the track for maximum speed and safety—ensuring smooth, thrilling curves!

5. Physics & Sports

1. Projectile motion: Calculus is used to model the arc of a football, basketball, or even a rocket.

2. Fitness trackers: These devices calculate energy used by measuring acceleration, which comes from derivatives of your movement data.

Why is Calculus Hard?

• It introduces abstract concepts like limits, infinity, and instantaneous change.

• You also learn new symbols and notation:
  For example:

1. 1. 1. dy/dx → the derivative

      2. ∫ → the integral

      3. lim → the limit

• But once you get the hang of visualising curves and thinking about movement or accumulation, it starts to make sense.

Newton vs. Leibniz – The Great Calculus Feud

• • Isaac Newton (England) developed calculus to explain motion and gravity.

  • Gottfried Wilhelm Leibniz (Germany) developed it independently and published first—with a beautiful, clean notation system.

  • Newton accused Leibniz of copying, and their supporters clashed in academic circles for years.

  • Today, both are recognised:

    1. Newton for the ideas

    2. Leibniz for the notation we still use today (like ∫ and d)

Key Calculus Concepts – A Glossary

• Derivative: Measures how something is changing right now. Think of it as the slope of a curve at a single point.

• Integral: Measures how much has built up over time. Think of it as the area under a curve.

• Limit: The value a function approaches as you get close to a certain point.

• Chain Rule: A way to find the derivative of a function inside another function.

• The Fundamental Theorem of Calculus: Shows how derivatives and integrals are linked—two sides of the same coin.

Why Calculus Still Matters

• It’s used in climate science, medical research, AI, finance, space travel, and engineering.

• It helps us understand motion, growth, and change in a world that’s always moving.

• Even if you don’t become a mathematician, it trains you to solve complex problems, think critically, and connect ideas.

Reference List

 Khan Academy – Calculus Course
https://www.khanacademy.org/math/calculus-1

3Blue1Brown – The Essence of Calculus (YouTube Series)
https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

WolframAlpha – Solve Derivatives and Integrals
https://www.wolframalpha.com

Stewart, James – Calculus: Early Transcendentals
[Available from many academic bookstores and libraries]

Wikipedia – Newton vs. Leibniz Calculus Controversy
https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy