AP PreCalculus Practice Quiz: Vector-Valued Functions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

A particle's position in the xy-plane is given by the parametric equations x(t) = 5t and y(t) = cos(t). Which of the following vector-valued functions represents the position of the particle?

A)p⃗(t) = 5t i⃗ + cos(t) j⃗B)p⃗(t) = 5 i⃗ - sin(t) j⃗C)p⃗(t) = cos(t) i⃗ + 5t j⃗D)p⃗(t) = ⟨5, -sin(t)⟩

All Questions (9)

A particle's position in the xy-plane is given by the parametric equations x(t) = 5t and y(t) = cos(t). Which of the following vector-valued functions represents the position of the particle?

A) p⃗(t) = 5t i⃗ + cos(t) j⃗

B) p⃗(t) = 5 i⃗ - sin(t) j⃗

C) p⃗(t) = cos(t) i⃗ + 5t j⃗

D) p⃗(t) = ⟨5, -sin(t)⟩

Correct Answer: A

The position of a particle given by parametric equations x(t) and y(t) can be expressed as the vector-valued function p⃗(t) = x(t)i⃗ + y(t)j⃗. Substituting the given equations, we get p⃗(t) = 5ti⃗ + cos(t)j⃗.

The position of a particle moving in a plane is described by the vector-valued function p⃗(t) = (t^2 + 1)i⃗ + (ln(t))j⃗. What is the particle's vertical position, y(t)?

A) t^2 + 1

B) 2t

C) ln(t)

D) 1/t

Correct Answer: C

The position vector is given in the form p⃗(t) = x(t)i⃗ + y(t)j⃗. By comparing this general form to the given function, the vertical component y(t), which is the coefficient of j⃗, is ln(t).

The position of a particle is given by the vector-valued function p⃗(t) = ⟨e^(2t), sin(t)⟩. Which of the following is the velocity vector of the particle at time t?

A) v⃗(t) = ⟨(1/2)e^(2t), -cos(t)⟩

B) v⃗(t) = ⟨e^(2t), cos(t)⟩

C) v⃗(t) = ⟨2e^(2t), cos(t)⟩

D) v⃗(t) = ⟨e^(2t), -cos(t)⟩

Correct Answer: C

The velocity vector v⃗(t) is the derivative of the position vector p⃗(t). We differentiate each component: the derivative of x(t) = e^(2t) is x'(t) = 2e^(2t), and the derivative of y(t) = sin(t) is y'(t) = cos(t). Therefore, v⃗(t) = ⟨2e^(2t), cos(t)⟩.

If the position of a particle is given by the parametric function f(t) = (x(t), y(t)), which of the following represents the velocity vector of the particle?

A) v⃗(t) = ⟨x(t), y(t)⟩

B) v⃗(t) = ⟨x'(t), y'(t)⟩

C) v⃗(t) = x'(t) + y'(t)

D) v⃗(t) = √(x(t)² + y(t)²)

Correct Answer: B

The content states that the velocity of a particle moving in a plane is expressed by the vector-valued function v⃗(t) = ⟨x'(t), y'(t)⟩, where x'(t) and y'(t) are the derivatives of the position components.

A particle moves in the xy-plane with its position given by the vector p⃗(t) = (t³)i⃗ + (4t)j⃗. What is the velocity vector of the particle at t = 2?

A) ⟨8, 8⟩

B) ⟨6, 4⟩

C) ⟨12, 4⟩

D) ⟨8, 4⟩

Correct Answer: C

First, find the general velocity vector by differentiating the position vector: v⃗(t) = ⟨d/dt(t³), d/dt(4t)⟩ = ⟨3t², 4⟩. Then, substitute t = 2 into the velocity vector: v⃗(2) = ⟨3(2)², 4⟩ = ⟨3(4), 4⟩ = ⟨12, 4⟩.

The velocity of a particle moving in a plane is given by the vector-valued function v⃗(t) = ⟨6, 8t⟩. Which of the following expressions represents the speed of the particle at time t?

A) 6 + 8t

B) √(36 + 64t²)

C) ⟨0, 8⟩

D) 36 + 64t²

Correct Answer: B

The speed is the magnitude of the velocity vector. The magnitude of a vector ⟨a, b⟩ is √(a² + b²). For the velocity vector v⃗(t) = ⟨6, 8t⟩, the speed is |v⃗(t)| = √(6² + (8t)²) = √(36 + 64t²).

The position of a particle is described by the vector-valued function p⃗(t) = ⟨3t, 4t⟩. What is the speed of the particle at t = 2?

A) ⟨3, 4⟩

B) 5

C) 7

D) 25

Correct Answer: B

First, find the velocity vector by differentiating the position vector: v⃗(t) = ⟨d/dt(3t), d/dt(4t)⟩ = ⟨3, 4⟩. The velocity vector is constant. The speed is the magnitude of the velocity vector: Speed = |v⃗(t)| = √(3² + 4²) = √(9 + 16) = √25 = 5. Since the speed is constant, it is 5 at all times, including t=2.

Which of the following correctly defines the speed of a particle whose motion is described by the position vector p⃗(t) = x(t)i⃗ + y(t)j⃗?

A) The derivative of the position vector, ⟨x'(t), y'(t)⟩.

B) The magnitude of the position vector, √(x(t)² + y(t)²).

C) The magnitude of the velocity vector, √(x'(t)² + y'(t)²).

D) The vector sum of the derivatives, x'(t)i⃗ + y'(t)j⃗.

Correct Answer: C

The content states that the speed of the particle is the magnitude of the velocity vector. The velocity vector is v⃗(t) = ⟨x'(t), y'(t)⟩. Therefore, the speed is its magnitude, |v⃗(t)| = √(x'(t)² + y'(t)²).

A particle's position is given by p⃗(t) = ⟨cos(t), sin(t)⟩. Which statement is true about the particle's motion?

A) The particle's speed is 1.

B) The particle's velocity is 1.

C) The particle is stationary.

D) The particle's speed is t.

Correct Answer: A

First, find the velocity vector: v⃗(t) = ⟨d/dt(cos(t)), d/dt(sin(t))⟩ = ⟨-sin(t), cos(t)⟩. Next, find the speed, which is the magnitude of the velocity vector: Speed = |v⃗(t)| = √((-sin(t))² + (cos(t))²) = √(sin²(t) + cos²(t)) = √1 = 1. The speed is constant and equal to 1.