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Topic 07 · 4 lessons

Calculus

Calculus studies things that change. Speed is the rate at which distance changes; acceleration is the rate at which speed changes. Calculus gives us precise tools for rates of change (differentiation) and for adding up infinitely many tiny pieces (integration).

These lessons assume you are comfortable with algebra and graphs. We build from the idea of a limit to the core techniques used in physics, engineering, economics and data science.

Lessons in this topic

  1. 7.1Understanding LimitsA limit describes where a function is heading, even if it never quite gets there or is undefined at that exact point.
  2. 7.2DerivativesThe derivative measures instantaneous rate of change: the gradient of a curve at a single point.
  3. 7.3IntegrationIntegration adds up infinitely many thin slices to find a total. It also undoes differentiation.
  4. 7.4The Chain RuleTo differentiate a function inside a function, differentiate the outside, keep the inside, then multiply by the derivative of the inside.

The big ideas

a)Zoom in until it looks straight
Zoom in far enough on a smooth curve and it looks like a straight line. The slope of that line is the derivative.
b)Limits handle "almost"
A limit describes the value a function approaches, even when we cannot plug that point in directly.
c)Add up slices
An integral finds a total, such as an area or a distance, by adding up infinitely many very thin slices.
d)Two sides of one coin
The Fundamental Theorem of Calculus says differentiation and integration undo each other.

Worked example

How fast is the ball falling?

A ball has fallen d=5t2d = 5t^2 metres after tt seconds. How fast is it moving at t=3t = 3?

  1. Speed is the rate of change of distance, so differentiate dd with respect to tt.
  2. Using the power rule: ddt(5t2)=10t\dfrac{d}{dt}(5t^2) = 10t.
  3. Substitute t=3t = 3: 10×3=3010 \times 3 = 30.

Answer: The ball is falling at 3030 metres per second.

Where you will meet it

Calculus predicts planetary orbits, models the spread of diseases, optimises delivery routes, trains machine-learning models and works out how much material a container really needs.

The wider field

Calculus also covers these ideas, which build on the lessons above:

  • Continuity
  • Product and quotient rules
  • Maxima and minima
  • Area under curves
  • Differential equations
  • Series