Vector Valued Functions and Their Differentiations

Maths, HSSC-Punjab Second Year. This chapter is split into 31 topics and has 849 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Derivative of a vector function: polynomial and power components 30 questions
2. Velocity vector of a particle from its position vector 30 questions
3. Constructing a vector function and finding its domain: steps 22 questions
4. Direction of motion: the unit tangent vector 30 questions
5. Derivative of a vector function: inverse trigonometric components 28 questions
6. Circular motion: velocity, acceleration and centripetal force 30 questions
7. Helical motion in three dimensions (set 2) 21 questions
8. Derivative of a vector function: exponential and logarithmic components 30 questions
9. Speed as the magnitude of the velocity 24 questions
10. Acceleration vector from the position vector 30 questions
11. Derivative of a vector function: trigonometric components 30 questions
12. Domain of a vector valued function 30 questions
13. Velocity, acceleration, speed and direction: definitions 21 questions
14. Evaluating a vector valued function at given values 30 questions
15. Applications: conveyor belt, comet, robotic arm and sensors (set 2) 23 questions
16. Derivative of a vector function at a given value of t (algebraic) 30 questions
17. Derivative of a vector valued function: definition and rules 21 questions
18. Curves traced by vector functions 30 questions
19. Applications: conveyor belt, comet, robotic arm and sensors (set 1) 23 questions
20. Constructing a vector function from component functions 28 questions
21. Helical motion in three dimensions (set 1) 20 questions
22. Helical motion in three dimensions (set 3) 28 questions
23. Projectile motion with a vector position function (set 2) 29 questions
24. Second derivative of a vector function 30 questions
25. Identifying scalar and vector valued functions 30 questions
26. Projectile motion with a vector position function (set 1) 29 questions
27. Derivative of a vector function: products, quotients and compositions 30 questions
28. Speeding up or slowing down (sign of v dot a) 30 questions
29. Second derivative of a vector function at a given value of t 30 questions
30. Derivative of a vector function at a given value of t (trigonometric) 30 questions
31. Scalar and vector valued functions: definitions 22 questions

Sample questions: Vector Valued Functions and Their Differentiations

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. The domain of r(t) = ⟨√(9 − t²), √(4 − t²), 1/(t − 3)⟩ is:

• A. (−∞, 1) ∪ (1, ∞)
• B. [−2, 2](correct answer)
• C. all real numbers
• D. [−3, 3]

Answer: B.The domain is the intersection of the domains of the components: for √(9 − t²): [−3, 3]; for √(4 − t²): [−2, 2]; for 1/(t − 3): (−∞, 3) ∪ (3, ∞). Intersection: [−2, 2].

Q2. The domain of r(t) = (1/(t − 1)) i + (√(25 − t²)) j + (ln(t + 2)) k is:

• A. (−∞, 1) ∪ (1, ∞)
• B. [−2, 5]
• C. (−2, 1) ∪ (1, 5](correct answer)
• D. [−5, 5]

Answer: C.The domain is the intersection of the domains of the components: for 1/(t − 1): (−∞, 1) ∪ (1, ∞); for √(25 − t²): [−5, 5]; for ln(t + 2): (−2, ∞). Intersection: (−2, 1) ∪ (1, 5].

Q3. The domain of r(t) = ⟨1/t, 2t + 1, 1/(t² − 4)⟩ is:

• A. (−∞, 0) ∪ (0, ∞)
• B. (−∞, −2) ∪ (−2, 0) ∪ (0, 2) ∪ (2, ∞)(correct answer)
• C. all real numbers
• D. (−∞, 1) ∪ (1, ∞)

Answer: B.The domain is the intersection of the domains of the components: for 1/t: (−∞, 0) ∪ (0, ∞); for 2t + 1: all real numbers; for 1/(t² − 4): (−∞, −2) ∪ (−2, 2) ∪ (2, ∞). Intersection: (−∞, −2) ∪ (−2, 0) ∪ (0, 2) ∪ (2, ∞).

Q4. The domain of r(t) = (1/t) i + (1/(t − 1)) j is:

• A. (−∞, 0) ∪ (0, ∞)
• B. all real numbers
• C. (−∞, 1) ∪ (1, ∞)
• D. (−∞, 0) ∪ (0, 1) ∪ (1, ∞)(correct answer)

Answer: D.The domain is the intersection of the domains of the components: for 1/t: (−∞, 0) ∪ (0, ∞); for 1/(t − 1): (−∞, 1) ∪ (1, ∞). Intersection: (−∞, 0) ∪ (0, 1) ∪ (1, ∞).

Q5. The domain of r(t) = ⟨1/(t² − 4), 1/(t − 3)⟩ is:

• A. (−∞, 3) ∪ (3, ∞)
• B. (−∞, 1) ∪ (1, ∞)
• C. (−∞, −2) ∪ (−2, 2) ∪ (2, 3) ∪ (3, ∞)(correct answer)
• D. all real numbers

Answer: C.The domain is the intersection of the domains of the components: for 1/(t² − 4): (−∞, −2) ∪ (−2, 2) ∪ (2, ∞); for 1/(t − 3): (−∞, 3) ∪ (3, ∞). Intersection: (−∞, −2) ∪ (−2, 2) ∪ (2, 3) ∪ (3, ∞).

Q6. The domain of r(t) = ⟨1/(t − 1), √t, ln(t + 2)⟩ is:

• A. [0, 1) ∪ (1, ∞)(correct answer)
• B. all real numbers
• C. (−∞, 1) ∪ (1, ∞)
• D. [0, ∞)

Answer: A.The domain is the intersection of the domains of the components: for 1/(t − 1): (−∞, 1) ∪ (1, ∞); for √t: [0, ∞); for ln(t + 2): (−2, ∞). Intersection: [0, 1) ∪ (1, ∞).

Q7. The domain of r(t) = ⟨√(9 − t²), 2t + 1, 1/(t − 1)⟩ is:

• A. all real numbers
• B. [−3, 3]
• C. [−3, 1) ∪ (1, 3](correct answer)
• D. (−∞, 1) ∪ (1, ∞)

Answer: C.The domain is the intersection of the domains of the components: for √(9 − t²): [−3, 3]; for 2t + 1: all real numbers; for 1/(t − 1): (−∞, 1) ∪ (1, ∞). Intersection: [−3, 1) ∪ (1, 3].

Q8. If r(t) = ⟨(t² + 1)/(sin t), t/(t² + 1)⟩, then r′(π/6) equals:

• A. ⟨2π/3 − √3(36 + π²)/18, 6π/(36 + π²)⟩
• B. ⟨2π/3 − √3(36 + π²)/18, 36(36 − π²)/(1296 + 72π² + π⁴)⟩(correct answer)
• C. ⟨4π/3 − √3(36 + π²)/9, 72(36 − π²)/(1296 + 72π² + π⁴)⟩
• D. ⟨2π/3 − √3(36 + π²)/18, 36(π² − 36)/(1296 + 72π² + π⁴)⟩

Answer: B.Differentiate each component: r′(t) = ⟨2t/(sin t) − (t² + 1) cos t/(sin²t), −2t²/(t² + 1)² + 1/(t² + 1)⟩, then put t = π/6: ⟨2π/3 − √3(36 + π²)/18, 36(36 − π²)/(1296 + 72π² + π⁴)⟩.

Q9. A particle moves with position vector r(t) = ⟨ln(t² + 4), 2t² − 5, 4 sin 5t⟩. Its acceleration at t = π/6 is:

• A. ⟨ln((144 + π²)/36), 4, −50⟩
• B. ⟨72(144 − π²)/(20736 + 288π² + π⁴), 4, −50⟩(correct answer)
• C. ⟨72(144 − π²)/(20736 + 288π² + π⁴), −4, −50⟩
• D. ⟨72(π² − 144)/(20736 + 288π² + π⁴), 4, −50⟩

Answer: B.Acceleration is the derivative of velocity, a(t) = r″(t) = ⟨−4t²/(t² + 4)² + 2/(t² + 4), 4, −100 sin 5t⟩; at t = π/6 it equals ⟨72(144 − π²)/(20736 + 288π² + π⁴), 4, −50⟩.

Q10. A particle has position r(t) = ⟨ln(t² + 4), 2t⁴ − t⟩. At t = π/6 it is:

• A. speeding up(correct answer)
• B. moving at constant speed
• C. slowing down
• D. turning around with zero speed

Answer: A.v(π/6) = ⟨12π/(144 + π²), π³/27 − 1⟩ and a(π/6) = ⟨72(144 − π²)/(20736 + 288π² + π⁴), 2π²/3⟩, so v·a = 2(−(27 − π³)(144 + π²)(20736 + 288π² + π⁴)π + 5038848 − 34992π²)π/(81(144 + π²)(20736 + 288π² + π⁴)) > 0: the speed is increasing.