What a Vector-Valued Function Is
A vector-valued function describes a point whose position depends on time. Instead of giving one output for each input, it gives a vector containing the point's horizontal and vertical coordinates.
For a particle moving in the plane, we commonly write its position as:
Here, gives the particle's horizontal position and gives its vertical position at time .
A vector-valued function is really two ordinary functions working together. That means differentiation is much less mysterious than it may initially look: differentiate each component separately.
How to Differentiate a Vector-Valued Function
Suppose:
Then the derivative is obtained by differentiating the component and the component independently:
The same rule works for vector-valued functions written using unit vectors:
Differentiate the coefficient of and the coefficient of separately. The unit vectors themselves do not change because they are constant.
The First Derivative: Velocity
In AP Calculus BC, vector-valued functions are often used to describe motion. If represents the position of a particle at time , then its derivative is the velocity vector.
The horizontal component of velocity tells you how quickly the particle's position is changing. The vertical component tells you how quickly its position is changing.
The particle's speed is the magnitude of its velocity vector:
Notice the distinction: velocity is a vector, while speed is a scalar. Velocity tells you both how fast the particle is moving and the direction in which it is moving.
The Second Derivative: Acceleration
Differentiate the velocity vector one more time and you obtain the acceleration vector.
Just as with the first derivative, differentiate each component separately:
So the standard motion framework is:
Worked Examples
Let . Find .
Let . Find the velocity vector and the speed at .
The position of a particle is . Find its velocity and acceleration vectors.
Let . Find the velocity vector.
Position, Velocity, and Acceleration
It helps to think of the three vectors as a chain:
For example, if you are given a position function, differentiate once to get velocity and twice to get acceleration. If you are given velocity, integrate to recover position only when enough initial information is supplied.
What Does a Velocity Component Tell You?
Suppose the velocity is . Then:
- means the particle's horizontal position is increasing.
- means the particle's vertical position is decreasing.
- The particle is moving diagonally because both velocity components are nonzero.
The signs of the velocity components therefore tell you the direction of motion in the coordinate plane.
Common Mistakes
A vector-valued function has multiple components. Differentiate each component separately. Do not try to apply one derivative rule to the entire vector at once.
Velocity is the vector . Speed is its magnitude, . If an AP question asks for speed, you must take the magnitude.
Each component is an ordinary function. If one component contains something like , you still need the chain rule when differentiating that component.
Read the wording carefully. Position is , velocity is its first derivative, and acceleration is its second derivative.
Practice Problems
Let . Find .
Show answer
A particle has position . Find its velocity at .
Show answer
For the particle in Question 2, find its speed at .
Show answer
Let . Find the acceleration vector .
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon