12 min read

Introduction

Rates of change and accumulation — how fast something grows, and how much builds up.

How Fast Is the Tank Filling?

A tank is filling. At one moment the water is rising quickly; later it rises more slowly. A single average — “litres per minute for the whole hour” — hides that story. Differentiation finds the rate of change at an instant (the gradient of a curve). Integration adds change back up to find a total (area under a curve). Calculus is those two ideas, sharpened into rules you can compute.

Example 1 — Average vs instant

If distance rises from 0 to 100 km in 2 hours, average speed is 50 km/h. That does not tell you the speedometer reading at 30 minutes. Differentiation is the tool for that instantaneous reading on a smooth curve.

You will build gradients from first principles, learn differentiation rules, interpret stationary points, then reverse the process with integration and area under a curve.

Practice

You need the gradient of a curve at one point. Which tool do you want?

More lessons in Calculus · Next: Rate of Change

12 min read

Rates of change and accumulation — how fast something grows, and how much builds up.

A tank is filling. At one moment the water is rising quickly; later it rises more slowly. A single average — “litres per minute for the whole hour” — hides that story. Differentiation finds the rate of change at an instant (the gradient of a curve). Integration adds change back up to find a total (area under a curve). Calculus is those two ideas, sharpened into rules you can compute.

Example 1 — Average vs instant

If distance rises from 0 to 100 km in 2 hours, average speed is 50 km/h. That does not tell you the speedometer reading at 30 minutes. Differentiation is the tool for that instantaneous reading on a smooth curve.

You will build gradients from first principles, learn differentiation rules, interpret stationary points, then reverse the process with integration and area under a curve.

Practice

You need the gradient of a curve at one point. Which tool do you want?

More lessons in Calculus · Next: Rate of Change