Parametric & Polar: every key term you need (+ practice quiz)

More AP Calculus BC guides

• Limits & Continuity
• Derivatives
• Applications of Derivatives
• Integration
• Applications of Integrals

Parametric Equations

Curve defined by x=f(t), y=g(t) where t is parameter (often time). Single parameter traces 2D curve.

Parametric Derivatives

dy/dx = (dy/dt)/(dx/dt). Second derivative d²y/dx² = d/dt(dy/dx) / (dx/dt). Used for finding slopes, tangent lines.

Speed & Velocity (Parametric)

Velocity vector = (dx/dt, dy/dt). Speed = √[(dx/dt)² + (dy/dt)²]. Arc length = ∫√[(dx/dt)² + (dy/dt)²] dt.

Concavity (Parametric)

Curve concave up if d²y/dx² > 0. Requires careful handling since x increases nonlinearly.

Eliminating Parameter

Solve one equation for t, substitute into other to get y=f(x). Useful for identifying curve type (circle, ellipse, etc.).

Polar Coordinates

Point location: (r, θ) where r = distance from origin, θ = angle from positive x-axis. Convert: x=r cos θ, y=r sin θ.

Polar to Rectangular

x=r cos θ, y=r sin θ, r²=x²+y², tan θ=y/x. Convert when needed for integration or analysis.

Rectangular to Polar

r=√(x²+y²), θ=arctan(y/x). Important for curves like spirals, roses, lemniscates.

Polar Derivatives

dy/dx = (r' sin θ + r cos θ)/(r' cos θ - r sin θ) where r' = dr/dθ. Used for tangent line slopes.

Polar Curve Types

r=a (circle), r=a cos θ (circle), r=a sin nθ (rose), r=ae^(bθ) (spiral), r²=a² cos 2θ (lemniscate).

Polar Area

Area = ½∫r² dθ from θ₁ to θ₂. Accounts for sector area swept by radius vector.

Unit 9 Summary

Parametric equations describe motion; dy/dx = (dy/dt)/(dx/dt). Polar coordinates useful for circular/radial motion; area = ½∫r² dθ.

Parametric Derivative

dy/dx = (dy/dt)/(dx/dt), defined wherever dx/dt ≠ 0. The parameter cancels, so the slope depends only on the ratio of the component rates.

Vertical and Horizontal Tangents

A parametric curve has a horizontal tangent where dy/dt = 0 with dx/dt ≠ 0, and a vertical tangent where dx/dt = 0 with dy/dt ≠ 0.

Parametric Concavity

d²y/dx² = (d/dt of dy/dx) ÷ (dx/dt). The final division by dx/dt converts a t-rate into an x-rate.

Vector-Valued Position

r(t) = ⟨x(t), y(t)⟩ has velocity ⟨x'(t), y'(t)⟩ and acceleration ⟨x''(t), y''(t)⟩; each component is differentiated separately.

Speed and Direction

Speed is the magnitude of velocity; direction of motion is given by the signs of the components. A particle at rest has both components zero.

Polar to Cartesian

x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x. Conversions let you check a polar answer against familiar Cartesian curves.

Slope of a Polar Curve

dy/dx = (r' sin θ + r cos θ)/(r' cos θ - r sin θ), obtained by differentiating the parametric forms x=r(θ)cos θ and y=r(θ)sin θ.

Polar Area Formula

A = (1/2)∫(α to β) r² dθ. The one-half comes from the area of a circular sector, not from averaging.

Area Between Polar Curves

A = (1/2)∫(r_outer² - r_inner²)dθ over the θ-range where that ordering holds; the intersection angles set the limits.

Limaçon Classification

r = a + b cos θ is a cardioid when a=b, has an inner loop when a<b, and is dimpled or convex when a>b.

Rose Curve Petals

r = a cos(nθ) has n petals when n is odd and 2n petals when n is even; each petal is traced over a θ-interval of π/n.

Polar Symmetry Tests

Replacing θ with -θ tests symmetry about the polar axis, θ with π-θ tests about the vertical line, and r with -r tests about the pole.

Negative r Values

A negative r plots the point in the opposite direction from θ, so the same point has many (r,θ) representations — this complicates finding intersections.

Polar Intersections

Solving r1 = r2 can miss intersections that occur at different θ values or at the pole; always check the origin separately.

Parametric Arc Length

L = ∫√((dx/dt)² + (dy/dt)²)dt, provided the interval traces the curve exactly once; retracing double-counts the length.

Polar Arc Length

L = ∫√(r² + (dr/dθ)²)dθ, the polar analogue where the radial and angular contributions add in quadrature.

Eliminating the Parameter

Solve one equation for t and substitute, or use an identity such as cos²+sin²=1 for trigonometric parametrizations, then note any domain restrictions.

Direction of Tracing

Increasing t may trace a curve clockwise or counterclockwise; test a few t values, since orientation affects motion questions but not the shape.

Cycloid

x = a(t - sin t), y = a(1 - cos t) has cusps where the tracing point touches the ground at t = 2πk, and dx/dt = dy/dt = 0 there.

Polar Tangent at the Pole

If r(θ0)=0 and r'(θ0) ≠ 0, the line θ = θ0 is tangent to the curve at the pole, which is the fastest way to sketch petal directions.