Vector-Valued Functions: An Introduction

What Is a Vector-Valued Function?

In ordinary calculus, a function such as assigns a single number to each input . A vector-valued function assigns a vector to each input instead.

For AP Calculus BC, the most important vector-valued functions have two components:

Here, and are ordinary functions of . Together, they specify both the horizontal and vertical coordinates of a point.

That makes vector-valued functions especially useful for describing motion in the plane. Instead of saying where an object is with one number, we can describe its entire position using an -coordinate and a -coordinate.

Think of It as Two Functions at Once

The easiest way to understand a vector-valued function is to separate it into its two components:

The first component tells you the object's horizontal position. The second tells you its vertical position.

For example, if

then at , the object is at the point .

As changes, the point moves. The collection of points traced out by the function is called its trajectory.

Position, Velocity, and Acceleration

This is where vector-valued functions become especially important in AP Calculus BC. If the position of a particle is given by

then its velocity is the derivative of the position:

and its acceleration is the derivative of velocity:

In component form:

So the basic pattern is exactly what you already know from one-dimensional motion:

The difference is that now each quantity has both an component and a component.

Worked Examples

How Vector Derivatives Work

There is nothing mysterious about differentiating a vector-valued function. Simply differentiate each component separately.

The same idea works for higher derivatives:

This component-by-component approach is one of the most useful habits to develop for AP Calculus BC. If a vector expression looks intimidating, separate it into its horizontal and vertical components and differentiate each one independently.

Position vs. Velocity vs. Speed

These three quantities are closely related, but they are not interchangeable.

• Position tells you where the particle is.
• Velocity tells you how position is changing and includes direction.
• Speed is the magnitude of velocity and tells you how fast the particle is moving.
• Acceleration tells you how velocity is changing.

Notice the important distinction: velocity is a vector, while speed is a scalar. Therefore, velocity can be positive, negative, or zero in either component, while speed can never be negative.

Common Mistakes

Practice Problems