1 · Parametric equations — a parameter drives the point
A parametric curve is given by x = f(t), y = g(t). As t runs, the point traces a path; t is usually time but can be any variable. Eliminating t recovers an ordinary equation when possible, but the parametric form keeps direction and pacing, which the Cartesian equation alone throws away. One curve can have many parametrizations — speed differs, shape does not.
2 · Derivatives of parametric curves
By the chain rule, dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. For second derivatives, differentiate dy/dx with respect to t and divide by dx/dt again: d²y/dx² = [d/dt(dy/dx)] ÷ (dx/dt). Horizontal tangents occur where dy/dt = 0 (and dx/dt ≠ 0); vertical tangents where dx/dt = 0 (and dy/dt ≠ 0).
Step 1: dx/dt = 3t² − 3, dy/dt = 2t.
Step 2: dy/dx = 2t/(3t² − 3). At t = 2: dy/dx = 4/(12 − 3) = 4/9.
Step 3: the point is x = 8 − 6 = 2, y = 4. Tangent: y − 4 = (4/9)(x − 2).
3 · Speed
Velocity is the vector (dx/dt, dy/dt); speed is its magnitude: √((dx/dt)² + (dy/dt)²). This is exactly the integrand for parametric arc length, L = ∫ √((x′)² + (y′)²) dt — so speed also measures distance traveled along the curve.
Step 1: dx/dt = −3sin t, dy/dt = 4cos t.
Step 2: speed = √(9sin²t + 16cos²t).
Step 3: at t = π/2: √(9·1 + 16·0) = 3. The ellipse is traversed faster and slower along its length.
4 · Polar coordinates — and conversions
Polar coordinates label a point (r, θ): distance r from the origin and angle θ from the positive x-axis. Conversions both ways: x = r cos θ, y = r sin θ, r² = x² + y², and tan θ = y/x (fix the quadrant). Polar tames spirals, circles through the origin and centered limaçons that look horrendous in Cartesian form.
5 · Sketching polar curves — cardioids, limaçons, roses
- Circles: r = a (centered at pole), r = 2a cos θ (diameter on x-axis), r = 2a sin θ (y-axis).
- Cardioids: r = a(1 ± cos θ) or a(1 ± sin θ) — heart-shaped, one cusp at the pole.
- Limaçons: r = a + b cos θ. a ≥ b: dimpled or convex; a < b: inner loop.
- Roses: r = a cos(nθ) — n petals for odd n, 2n petals for even n.
Sketch by plotting r at θ = 0, π/2, π, 3π/2 and using symmetry (θ → −θ: x-axis symmetry; θ → π − θ: y-axis; θ → θ + π: pole). Negative r flips the point through the origin — never discard those values.
6 · Tangents to polar curves
Treat r = r(θ) as parametric with parameter θ: x = r cos θ, y = r sin θ. Differentiate: dx/dθ = r′cos θ − r sin θ, dy/dθ = r′sin θ + r cos θ. Then dy/dx = (r′ sin θ + r cos θ)/(r′ cos θ − r sin θ). Horizontal tangent: numerator zero. Vertical tangent: denominator zero. At the pole (r = 0) check whether the curve passes through with a defined tangent direction.
7 · Area enclosed by a polar curve
A thin wedge of radius r(θ) and angle dθ has area ½r²dθ, so summing gives the master formula A = ½ ∫ₐᵇ r² dθ. Between two curves: A = ½ ∫(r_out² − r_in²) dθ. Find limits by solving r₁ = r₂ (where the curves meet) or by sweeping exactly one petal/loop.
Step 1: r = sin θ traces the full circle once as θ runs 0 → π.
Step 2: A = ½ ∫₀^π sin²θ dθ = ½ · π/2 = π/4.
Step 3: check — r = sin θ is a circle of diameter 1 sitting on the x-axis; area π/4. ✓
x = r cos θ · y = r sin θ · r² = x² + y² · tan θ = y/x
A = ½ ∫ₐᵇ r² dθ (polar area) · area between: ½ ∫(r_out² − r_in²) dθ
Rose r = a cos nθ: n petals (n odd), 2n petals (n even) · cardioid r = a(1 ± cos θ)
Polar tangent: dy/dx = (r′sin θ + r cos θ)/(r′cos θ − r sin θ)
Common mistakes
- Writing dy/dx = dy/dt instead of dividing by dx/dt.
- Thinking dx/dt = 0 gives a horizontal tangent — it is vertical.
- Converting with tan θ = y/x and ignoring the quadrant of θ.
- Integrating r² dθ over 0 to 2π for a rose and double-counting petals.
- Forgetting the factor ½ in A = ½ ∫ r² dθ.
- Discarding negative r values when sketching — they belong on the graph.