Unit 9 Test Review: Parametric and Polar Concepts

Parametric equations

Parametric equations express the coordinates of a curve as functions of a third variable, usually time (t), in the form x(t) and y(t).

Derivative for parametric equations

Use the formula: dy/dx = (dy/dt) / (dx/dt).

Second derivative for parametric equations

Use the formula: d²y/dx² = (d/dt(dy/dx)) / (dx/dt).

Concavity for parametric equations

Use d²y/dx²: If d²y/dx² > 0, the curve is concave up. If d²y/dx² < 0, the curve is concave down.

Horizontal or vertical tangent for parametric curves

Horizontal tangent: dy/dt = 0 and dx/dt ≠ 0. Vertical tangent: dx/dt = 0 and dy/dt ≠ 0.

Arc length of a parametric curve

L = ∫√((dx/dt)² + (dy/dt)²) dt.

Speed of a particle in parametric equations

Speed = √((dx/dt)² + (dy/dt)²).

Polar curve equation

A polar equation expresses a curve as r(θ), where r is the radius and θ is the angle.

Slope of a polar curve

Use the formula: dy/dx = (dr/dθ sin(θ) + r cos(θ)) / (dr/dθ cos(θ) - r sin(θ)).

Area enclosed by a polar curve

A = (1/2) ∫ r(θ)² dθ.

Points of intersection for polar curves

Set the equations equal to each other: r₁(θ) = r₂(θ) and solve for θ.

Convert from polar to rectangular coordinates

Use the formulas: x = r cos(θ), y = r sin(θ).

Convert from rectangular to polar coordinates

Use the formulas: r = √(x² + y²), θ = arctan(y/x).

Position vector

The position vector r(t) describes the location of a particle. Velocity: v(t) = dr/dt. Acceleration: a(t) = d²r/dt².

Dot product of two vectors

A · B = |A||B|cos(θ) where A and B are vectors.

Equation of a tangent line for parametric curves

1. Find dy/dx. 2. Use the point-slope formula: y - y₀ = m(x - x₀) where m = dy/dx at the given t.