Key Idea: A limit is the value a function heads towards. In AI you use it to get the gradient of a curve: chord gradients head towards the tangent gradient, which is the derivative.
Key facts
- Average gradient = chord gradient = (f(b)−f(a))/(b−a).
- The tangent gradient (derivative) is the limit of the chord gradient.
- The derivative is the instantaneous rate of change. Give its units.
- Treat limits intuitively. No algebraic limit evaluation is required.
Chords on f(x) = x² near x = 2
- One chord — From x = 2 to x = 2.1: (4.41 − 4)/0.1 = 4.1.
- Slide closer — Move the second point nearer to x = 2.
- The limit — The gradient approaches 4, the gradient of the curve at x = 2.
A limit is about where f(x) is heading near a point, not its value at the point.
Find the average gradient of f(x) = x² from x = 1 to x = 3. (9 − 1)/2 = 4.
As the two ends of a chord slide together, what does its gradient approach? The tangent gradient: the derivative.
Distance is in metres and time in seconds. What units does the rate of change have? Metres per second.
Chord gradients on y = x² near x = 2 go 4.1, 4.01, 4.001. What is the limit? 4.