Differentiation of Vector Functions

Maths, HSSC-KPK Second Year. This chapter is split into 28 topics and has 675 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Proofs and verification of vector differentiation rules 21 questions
2. Direction of motion and unit tangent vector 26 questions
3. Dot product of vector functions 28 questions
4. Scalar functions 24 questions
5. Limits of vector functions with indeterminate forms 26 questions
6. Evaluating vector functions 26 questions
7. Vector functions: definition and notation 21 questions
8. Rules of vector differentiation 24 questions
9. Limits of vector functions: definition and rules 20 questions
10. Velocity from a position vector 26 questions
11. Differentiability of vector functions 21 questions
12. Chain rule for vector functions 26 questions
13. Linearity and scalar multiple rules in practice 26 questions
14. Acceleration and speed from a position vector 26 questions
15. Continuity of vector functions: definition and theorem 21 questions
16. Derivatives of vector functions 26 questions
17. Cross product of vector functions 26 questions
18. Domain of vector functions 20 questions
19. Operations on vector functions: sum, difference, scalar multiple 26 questions
20. Points of continuity and discontinuity of vector functions 21 questions
21. Curves traced by vector functions 24 questions
22. Differentiating cross products 26 questions
23. Position vector, velocity and acceleration: definitions 21 questions
24. Differentiating dot products 26 questions
25. Second derivative and derivative values of vector functions 26 questions
26. Scalar functions from dot products and magnitudes 24 questions
27. Derivative of a vector function: definition 21 questions
28. Evaluating limits of vector functions 26 questions

Sample questions: Differentiation of Vector Functions

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

Q1. Example 4.2.2: the function F(t) = (sin t, (1 − t)⁻¹) is continuous for

• A. all real t except t = 1(correct answer)
• B. t > 0 and t ≠ 1
• C. all real t
• D. t > 1

Answer: A.sin t is continuous everywhere and (1 − t)⁻¹ is continuous for t ≠ 1 (the book's text restricts further to t > 0, which is not required for this pair of components).

Q2. If F(t) = (t + 3) i + (t³ − 3) j + (t² − t) k and h(t) = cos t, then d/dt (hF) =

• A. (−t sin t − 3sin t + cos t) i + (−t³ sin t + 3t² cos t + 3sin t) j + (−t² sin t + t sin t + 2t cos t − cos t) k(correct answer)
• B. −(sin t) i − (3t² sin t) j + (−2t sin t + sin t) k
• C. (−t sin t − 3sin t) i + (−t³ sin t + 3sin t) j + (−t² sin t + t sin t) k
• D. (−t sin t − 3sin t − cos t) i + (−t³ sin t − 3t² cos t + 3sin t) j + (−t² sin t + t sin t − 2t cos t + cos t) k

Answer: A.Scalar multiple rule: (hF)' = h'F + hF'. Computing: (−t sin t − 3sin t + cos t) i + (−t³ sin t + 3t² cos t + 3sin t) j + (−t² sin t + t sin t + 2t cos t − cos t) k.

Q3. If F(t) = (5t³ − 2) i + (2t − 4) j + (−2t³ − 1) k and h(t) = sin t, then d/dt (hF) =

• A. (5t³ cos t − 15t² sin t − 2cos t) i + (2t cos t − 2sin t − 4cos t) j + (−2t³ cos t + 6t² sin t − cos t) k
• B. (15t² sin t) i + (2sin t) j − (6t² sin t) k
• C. (5t³ cos t + 15t² sin t − 2cos t) i + (2t cos t + 2sin t − 4cos t) j + (−2t³ cos t − 6t² sin t − cos t) k(correct answer)
• D. (5t³ cos t − 2cos t) i + (2t cos t − 4cos t) j + (−2t³ cos t − cos t) k

Answer: C.Scalar multiple rule: (hF)' = h'F + hF'. Computing: (5t³ cos t + 15t² sin t − 2cos t) i + (2t cos t + 2sin t − 4cos t) j + (−2t³ cos t − 6t² sin t − cos t) k.

Q4. If F(t) = (5 − 3t) i + (t² − 2t) j + (t² − t) k and h(t) = cos t, then d/dt (hF) =

• A. (3t sin t − 5sin t + 3cos t) i + (−t² sin t + 2t sin t − 2t cos t + 2cos t) j + (−t² sin t + t sin t − 2t cos t + cos t) k
• B. −(3cos t) i + (2t cos t − 2cos t) j + (2t cos t − cos t) k
• C. (3t sin t − 5sin t − 3cos t) i + (−t² sin t + 2t sin t + 2t cos t − 2cos t) j + (−t² sin t + t sin t + 2t cos t − cos t) k(correct answer)
• D. (3t sin t − 5sin t) i + (−t² sin t + 2t sin t) j + (−t² sin t + t sin t) k

Answer: C.Scalar multiple rule: (hF)' = h'F + hF'. Computing: (3t sin t − 5sin t − 3cos t) i + (−t² sin t + 2t sin t + 2t cos t − 2cos t) j + (−t² sin t + t sin t + 2t cos t − cos t) k.

Q5. If F(t) = (4t³ − 3) i + (t² − 4) j + (2t² − 3t) k and G(t) = (5t³ − 4) i + (2cos t) j + (3t² − 3t) k, then d/dt (F · G) =

• A. 120t⁵ + 24t³ − 2t² sin t − 138t² + 4t cos t + 18t + 8sin t(correct answer)
• B. 180t⁴ + 24t² − 4t sin t − 30t + 9
• C. −120t⁵ − 24t³ + 2t² sin t + 138t² − 4t cos t − 18t − 8sin t
• D. 60t⁵ + 12t³ − 2t² sin t − 69t² + 9t + 8sin t

Answer: A.F · G = 20t⁶ + 6t⁴ − 46t³ + 2t² cos t + 9t² − 8cos t + 12; differentiating (or using F'·G + F·G') gives 120t⁵ + 24t³ − 2t² sin t − 138t² + 4t cos t + 18t + 8sin t.

Q6. If F(t) = (t² − 2t) i + (2t² − 5t) j + (2t² − 3t) k and G(t) = −2t i + (3cos t) j + (5t³ − 3) k, then d/dt (F · G) =

• A. 50t⁴ − 60t³ − 6t² sin t − 6t² + 15t sin t + 12t cos t − 4t − 15cos t + 9(correct answer)
• B. 20t⁴ − 15t³ − 4t² + 12t cos t − 8t − 15cos t + 9
• C. −50t⁴ + 60t³ + 6t² sin t + 6t² − 15t sin t − 12t cos t + 4t + 15cos t − 9
• D. −10t⁴ + 30t³ + 6t² sin t − 2t² − 15t sin t + 12t cos t − 12t − 15cos t + 9

Answer: A.F · G = 10t⁵ − 15t⁴ − 2t³ + 6t² cos t − 2t² − 15t cos t + 9t; differentiating (or using F'·G + F·G') gives 50t⁴ − 60t³ − 6t² sin t − 6t² + 15t sin t + 12t cos t − 4t − 15cos t + 9.

Q7. Let F(u) = (2u + 1) i + (u² − u) j + (cos u) k and h(t) = t² + 1. Then d/dt F(h(t)) =

• A. 4t i + (4t³ + 2t) j − (2t sin(t² + 1)) k(correct answer)
• B. 2i + (2t − 1) j − (sin t) k
• C. (4t³ + 6t) i + (2t⁵ + 2t³) j + (2t cos(t² + 1)) k
• D. (2t + 2) i + (2t² + 2t + 1) j + (2t − sin(t² + 1)) k

Answer: A.F(h(t)) = (2t² + 3) i + (−t² + (t² + 1)² − 1) j + (cos(t² + 1)) k; differentiating each component (chain rule) gives 4t i + (4t³ + 2t) j − (2t sin(t² + 1)) k.

Q8. Let F(u) = (sin u) i + u³ j + (cos u) k and h(t) = 2t + 1. Then d/dt F(h(t)) =

• A. (cos(2t + 1) + 2) i + (12t² + 12t + 5) j + (2 − sin(2t + 1)) k
• B. (2sin(2t + 1)) i + (16t³ + 24t² + 12t + 2) j + (2cos(2t + 1)) k
• C. (2cos(2t + 1)) i + (24t² + 24t + 6) j − (2sin(2t + 1)) k(correct answer)
• D. (cos(2t + 1)) i + (12t² + 12t + 3) j − (sin(2t + 1)) k

Answer: C.F(h(t)) = (sin(2t + 1)) i + (2t + 1)³ j + (cos(2t + 1)) k; differentiating each component (chain rule) gives (2cos(2t + 1)) i + (24t² + 24t + 6) j − (2sin(2t + 1)) k.

Q9. Let F(u) = (cos u) i + (cos u) j + (sin u) k and h(t) = 2t + 1. Then d/dt F(h(t)) =

• A. (2 − sin(2t + 1)) i + (2 − sin(2t + 1)) j + (cos(2t + 1) + 2) k
• B. −(2sin(2t + 1)) i − (2sin(2t + 1)) j + (2cos(2t + 1)) k(correct answer)
• C. (2cos(2t + 1)) i + (2cos(2t + 1)) j + (2sin(2t + 1)) k
• D. −(sin(2t + 1)) i − (sin(2t + 1)) j + (cos(2t + 1)) k

Answer: B.F(h(t)) = (cos(2t + 1)) i + (cos(2t + 1)) j + (sin(2t + 1)) k; differentiating each component (chain rule) gives −(2sin(2t + 1)) i − (2sin(2t + 1)) j + (2cos(2t + 1)) k.

Q10. Let F(u) = (u² − u) i + u² j + u³ k and h(t) = 2t + 1. Then d/dt F(h(t)) =

• A. (2t − 1) i + 2t j + 3t² k
• B. (8t² + 4t) i + (8t² + 8t + 2) j + (16t³ + 24t² + 12t + 2) k
• C. (8t + 2) i + (8t + 4) j + (24t² + 24t + 6) k(correct answer)
• D. (4t + 1) i + (4t + 2) j + (12t² + 12t + 3) k

Answer: C.F(h(t)) = (−2t + (2t + 1)² − 1) i + (2t + 1)² j + (2t + 1)³ k; differentiating each component (chain rule) gives (8t + 2) i + (8t + 4) j + (24t² + 24t + 6) k.