Parametric Equations, Polar Coordinates & Vector-Valued Functions Cheat Sheet

The core ideas of Parametric Equations, Polar Coordinates & Vector-Valued Functions distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Parametric Equations

x=f(t), y=g(t) defining a curve via parameter t.

Parametric Derivative

dy/dx = (dy/dt)/(dx/dt), dx/dt \!= 0.

Polar Coordinates

Point (r,theta): distance and angle from origin.

Polar Area

A = (1/2) int r^2 dtheta.

Arc Length

L = int sqrt((dx/dt)^2+(dy/dt)^2) dt.

Vector-Valued Function

r(t)=<x(t),y(t)>.

Speed

|v(t)| = sqrt(x_prime^2 + y_prime^2).

Key Terms at a Glance

Parametric Equations:x=f(t),y=g(t) describing curve via t.

Polar Coordinates:(r,theta): distance and angle from origin.

Vector-Valued Function:r(t)=<x(t),y(t)>: scalar to vector.

Cardioid:Heart-shaped: r=a(1+/-cos theta).

Rose Curve:r=a cos(n theta): petal loops.

Limacon:r=a+b cos theta. a=b:cardioid. a<b:loop. a>b:convex.

Lemniscate:Figure-eight: r^2=a^2 cos(2theta).

Cycloid:Rolling circle: x=a(t-sin t),y=a(1-cos t).

Speed:|v|=sqrt(x_prime^2+y_prime^2).

Displacement:r(b)-r(a).

Arc Length:int speed dt or int sqrt(r^2+r_prime^2) dtheta.

Equiangular Spiral:r=a e^(b theta), constant angle with radii.

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