6 min readβ’january 25, 2023

Sumi Vora

Jed Quiaoit

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.Β When we deal with parametrics in the context of motion, we express them as **vector-valued functions**. Vector-valued functions arenβt graphed with the points x and y like we are used to seeing. Instead, each βpointβ 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. π§

One of the key concepts students learn is the derivative of a function. When dealing with **Cartesian** **functions**, such as y = f(x), the derivative of the function gives us the slope of the tangent line at any point on the curve. This slope is often represented as dy/dx, and it gives us a measure of how much y is changing as x changes.

When we move to the realm of **vector-valued functions**, we are still looking at how much a variable is changing, but now we are looking at how much the position of a particle is changing in both the x and y direction. In this case, the position vector, represented by r(t), tells us the position of a particle at any given time t. The velocity vector, represented by v(t), tells us how fast the particle is moving in the x and y direction at any given time t. And the acceleration vector, represented by a(t), tells us how the velocity of the particle is changing at any given time t.

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. These vectors are related, and together they give us a complete picture of how the position of a particle is changing over time.

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 arenβt 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. π

A particle is moving in a two-dimensional plane, and its position as a function of time, t, is given by the vector-valued function r(t) = <t^3 - 9t, 3t^2 - 6t>. The particle's velocity at any time t is given by the derivative of the position vector with respect to time, and its acceleration at any time t is given by the derivative of the velocity vector with respect to time. βοΈ

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