Parametric & polar calculus
The idea
Some curves refuse to be graphs of y = f(x) — orbits, spirals, loops that cross themselves. Parametric equations handle them by letting a parameter t drive both coordinates, x = x(t) and y = y(t), like a timestamped flight path; polar coordinates handle them by locating points with a radius r and angle θ. Calculus transfers to both settings: you can still ask for slopes, arc lengths, and areas, and you already own every tool needed — the chain rule and the definite integral.
For parametric slope, the chain rule gives dy/dx = (dy/dt)/(dx/dt): the slope of the path is the vertical speed divided by the horizontal speed. For polar area, slices are thin pie wedges, giving area as the integral of r²/2 with respect to θ. The reliable misconception to disarm: dy/dx is not obtained by differentiating y(t) alone. Differentiating y(t) gives a rate per unit of t, not per unit of x; forgetting to divide by dx/dt is the most common parametric error.
Worked example
A curve is given by x = t², y = t³ − 3t. Find the slope dy/dx of the tangent line at t = 2, and the point where it touches the curve.
- Differentiate each coordinate with respect to the parameter: dx/dt = 2t and dy/dt = 3t² − 3.
- Form the slope as the ratio of rates: dy/dx = (3t² − 3)/(2t), valid wherever dx/dt ≠ 0.
- Evaluate at t = 2: dy/dx = (12 − 3)/4 = 9/4, so the curve climbs 9 vertical units for every 4 horizontal units there.
- Locate the point of tangency from the original equations: x = 2² = 4 and y = 8 − 6 = 2, so the tangent touches at (4, 2).
- Check the trouble spot the formula warns about: at t = 0, dx/dt = 0 while dy/dt = −3 ≠ 0, so the curve has a vertical tangent there — the slope formula failing is information, not breakdown.
Answer. At t = 2 the curve passes through (4, 2) with tangent slope dy/dx = 9/4.
Check your understanding
- Why does dividing dy/dt by dx/dt produce the slope of the path through the plane?
- What is the geometric meaning of points where dx/dt = 0 but dy/dt ≠ 0, and where both vanish at once?
- How does the wedge picture explain the r²/2 in the polar area formula, rather than just r?
- When a parametric curve crosses itself, what does that imply about tangent lines at the crossing point?
Build the foundations first
Parametric & polar calculus builds on these concepts. If any feel shaky, start there.