Curves from a moving point

Draw a curve by letting a point move with time, and read its velocity, speed and tangent. About 8 minutes, A-level Further Maths.

What you will do

1. A point that moves. Instead of writing y as a function of x, we let a point move: x = t, y = t. The parameter t is like time. The bold part is where the point has been so far; right now t = t.
2. Slide it round. Drag the red point round the curve (or use the t slider). Take it to the top, where t = /2 1.57.
3. Velocity. The orange arrow is the velocity: (x'(t),\ y'(t)), which here is (- t,\ t). It always points along the curve — that is the tangent direction.
4. Slope of the tangent. The slope of the tangent is dydx = y'(t)x'(t). At t = /4 the point is at 45° on the circle.
5. The cycloid. Now x = t - t, y = 1 - t: the path of a point on a rolling wheel. Press Trace the curve or drag the point along it.
6. Your turn. Everything is unlocked. Type your own x(t) and y(t), try the presets, use sliders with letters like a and b, and watch the velocity change.