← Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC)
Parametric Equations & Vector-Valued Functions (BC)
Parametric equations describe x and y separately in terms of a parameter t; derivatives dy/dx are found via dy/dt divided by dx/dt. Vector-valued functions extend this to describe motion in the plane.
Differentiating parametric curves
For x(t) and y(t), dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. This comes directly from the chain rule: dy/dt = (dy/dx)(dx/dt).
For a position vector r(t) = ⟨x(t), y(t)⟩, velocity is r'(t) = ⟨x'(t), y'(t)⟩ and speed is the magnitude |r'(t)| = √([x'(t)]² + [y'(t)]²).
Key terms (1)
- Parametric derivative
- dy/dx = (dy/dt)/(dx/dt)
Practice Quiz
Question 1 of 1Score so far: 0/0
x(t)=t², y(t)=t³. Find dy/dx.