9.4 Defining and Differentiating Vector-Valued Functions
1 min read•april 26, 2020
Parametrics and Motion
Because parametric functions are associated with time, they are also generally used to calculate motion and velocity, and the College Board usually uses parametrics in this context. When we deal with parametrics in the context of motion, we express them as vector-valued functions. Vector-valued functions aren’t graphed with the points x and y like we are used to seeing. Instead, each “point” on a vector-valued function is determined by a position vector (a vector that starts at the origin) that exists in the direction of the point.
Just like Cartesian functions, if we take the derivative of the position vector, we would get the velocity vector, and if we take the derivative of the velocity vector, we would get the acceleration vector. When we were taking the derivative of a parametric function to find dy/dx, we were trying to find the slope of the tangent line that was determined by both the x and y functions of the curve. However, when we are looking at vector-valued functions, we aren’t looking at the curve itself; we are looking at how much our particle is moving in the direction of x and how much it is moving in the direction of y. This means that when we are taking derivatives of vector-valued functions, we take the derivative of the components separately.