AP Calculus BC glossary

Vector-valued function

A vector-valued function assigns a vector to each value of a parameter, most often describing the position of a particle over time. Differentiating each component separately gives the velocity vector, and differentiating again gives acceleration.

r(t)=⟨x(t), y(t)⟩\mathbf{r}(t) = \langle x(t),\, y(t) \rangle

Everything is componentwise, which makes these easier than they look. The velocity is ⟨x′(t),y′(t)⟩\langle x'(t), y'(t) \rangle and the speed is its magnitude, (x′(t))2+(y′(t))2\sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2}.

Total distance travelled is the integral of speed over the time interval, while displacement is the vector difference between the ending and starting positions.

Speed is a scalar

Velocity is a vector with direction; speed is a single non-negative number. Questions asking when a particle is at rest want both velocity components to be zero at once.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)