Parametric Equations, Polar Coordinates, and Conic Sections

These flashcards cover key concepts regarding parametric equations, polar coordinates, and conic sections, aiding in study and understanding of the topics.

Parametric Equations

Equations that express a curve in terms of one or more parameters.

Derivative of a parametric curve

Calculated using the formula dy/dx = (dy/dt)/(dx/dt) for x = x(t) and y = y(t).

Arc Length Formula for Parametric Curves

The formula to calculate the arc length is s = ∫√((dx/dt)² + (dy/dt)²) dt.

Polar Coordinates

An alternative coordinate system for locating points in a plane using radius and angle.

Conversion to Polar Coordinates

To convert rectangular coordinates (x,y) to polar coordinates (r,θ), use r = √(x² + y²) and θ = tan⁻¹(y/x).

Symmetry in Polar Curves

Polar curves can exhibit symmetry through the pole, horizontal axis, or vertical axis.

Area in Polar Coordinates

The area A of a region defined by r = f(θ) is given by A = (1/2)∫[f(θ)]² dθ.

Eccentricity of Conic Sections

A measure of how much a conic section deviates from being circular; for ellipses e < 1, parabolas e = 1, and hyperbolas e > 1.

Standard Form of a Parabola

The equation of a vertical parabola is y = 4p(x-h)² + k.

Discriminant for Conic Identification

To identify a conic generated by Ax² + Bxy + Cy² + Dx + Ey + F = 0, compute D = B² - 4AC: if D > 0, ellipse; D = 0, parabola; D < 0, hyperbola.