9.4 Defining and Differentiating Vector-Valued Functions

What is a vector-valued function?

A vector-valued function gives the position of a point using component functions of a parameter t: r(t) = ⟨x(t), y(t)⟩. Each component is a function of t, tracing a path (curve) in the plane as t varies. These are closely related to parametric equations, just written with vector notation instead of separate x(t) and y(t) equations.

Differentiating vector-valued functions

To differentiate r(t) = ⟨x(t), y(t)⟩, differentiate each component separately: r′(t) = ⟨x′(t), y′(t)⟩. This derivative vector represents the velocity vector, tangent to the path at time t. Its direction shows the direction of motion, and its magnitude gives speed.

Speed and higher derivatives

Speed is the magnitude of the velocity vector: |r′(t)| = √[(x′(t))² + (y′(t))²]. Differentiating again gives the acceleration vector r″(t) = ⟨x″(t), y″(t)⟩. These vectors describe motion along a curve just as they do for parametric motion problems.

Key terms

Vector-valued function

A function r(t) = ⟨x(t), y(t)⟩ that gives position as a vector depending on parameter t.

r′(t)

The velocity vector, found by differentiating each component: ⟨x′(t), y′(t)⟩.

Speed formula

|r′(t)| = √[(x′(t))² + (y′(t))²], the magnitude of the velocity vector.

r″(t)

The acceleration vector, found by differentiating the velocity vector: ⟨x″(t), y″(t)⟩.

Direction of motion

Given by the direction of the velocity vector r′(t), tangent to the curve at time t.

Try an AP-style question

If r(t) = ⟨t², sin t⟩, what is r'(t)?

1. ⟨2t, cos t⟩
2. 2t + cos t
3. ⟨2t, −cos t⟩
4. ⟨t, cos t⟩