Worksheets · Calculus

Vector-valued functions worksheet

A vector-valued function is a curve with a clock on it. Calculus on r(t) is done one component at a time: differentiate each to get the velocity, integrate each (with a constant vector) to go back, and use the length of r′(t) for speed and arc length.

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Sigma Prep · sigmaprep.io/worksheets

Name

Vector-Valued Functions

Date Period

Answer each question. Give exact answers.

  1. Find r′(t) for r(t)=⟨3t3−3,3ln⁡t,4cos⁡2t⟩
  2. r′(t)=⟨−3t2,2e2t,−6t+1⟩ and r(0)=⟨0,1,0⟩. Find r(t).
  3. Find the unit tangent vector T(2) for r(t)=⟨t2−10t+2,−2t2+2t+4,−2t2+15t−3⟩.
  4. Find the length of r(t)=⟨t2,2t,ln⁡t⟩, 1≤t≤4.
  5. Find r′(t) for r(t)=⟨−t3−6,4ln⁡t,4t2−t⟩
  6. r′(t)=⟨5,3(t+1)et,et⟩ and r(0)=⟨5,0,1⟩. Find r(t).
  7. Find the unit tangent vector T(3) for r(t)=⟨t2−12t+2,t2+4,−t2+13t−3⟩.
  8. Find the length of r(t)=⟨etcos⁡t,etsin⁡t,et⟩, 0≤t≤ln⁡4.
  9. Find r′(t) for r(t)=⟨−3t3+5,sin⁡t,tet⟩
  10. r′(t)=⟨−3t2,2e2t,6t−3⟩ and r(0)=⟨−5,1,0⟩. Find r(t).
  11. Find the unit tangent vector T(1) for r(t)=⟨−t2+8t,−2t2+10t−2,2t2−11t+1⟩.
  12. Find the length of r(t)=⟨4cos⁡t,4sin⁡t,3t⟩, 0≤t≤2π.
  13. Find r′(t) for r(t)=⟨5t2−4t,sin⁡t,2ln⁡t⟩
  14. r′(t)=⟨−6,−6sin⁡2t,9cos⁡3t⟩ and r(0)=⟨0,3,0⟩. Find r(t).
  15. Find the unit tangent vector T(1) for r(t)=⟨−2t2+7t−3,t2+2t+1,−2t2+16t+4⟩.
  16. Find the length of r(t)=⟨3cos⁡t,3sin⁡t,4t⟩, 0≤t≤π/2.
  17. Find r′(t) for r(t)=⟨−5t2+5t,e−2t,−4t2⟩
  18. r′(t)=⟨−10t+1,− ⁣sin⁡t,−3t2⟩ and r(0)=⟨0,1,2⟩. Find r(t).
  19. Find the unit tangent vector T(2) for r(t)=⟨2t2−12t+3,2t2−12t+4,−2t2+t+4⟩.
  20. Find the length of r(t)=⟨2t,t2,ln⁡t⟩, 1≤t≤6.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Vector-Valued FunctionsAnswer keyVersion 1

Date Period

Answer each question. Give exact answers.

  1. Find r′(t) for r(t)=⟨3t3−3,3ln⁡t,4cos⁡2t⟩r′(t)=⟨9t2,3t,−8sin⁡2t⟩
  2. r′(t)=⟨−3t2,2e2t,−6t+1⟩ and r(0)=⟨0,1,0⟩. Find r(t).r(t)=⟨−t3,e2t,−3t2+t⟩
  3. Find the unit tangent vector T(2) for r(t)=⟨t2−10t+2,−2t2+2t+4,−2t2+15t−3⟩.T(2)=⟨−611,−611,711⟩
  4. Find the length of r(t)=⟨t2,2t,ln⁡t⟩, 1≤t≤4.15+ln⁡4
  5. Find r′(t) for r(t)=⟨−t3−6,4ln⁡t,4t2−t⟩r′(t)=⟨−3t2,4t,8t−1⟩
  6. r′(t)=⟨5,3(t+1)et,et⟩ and r(0)=⟨5,0,1⟩. Find r(t).r(t)=⟨5t+5,3tet,et⟩
  7. Find the unit tangent vector T(3) for r(t)=⟨t2−12t+2,t2+4,−t2+13t−3⟩.T(3)=⟨−611,611,711⟩
  8. Find the length of r(t)=⟨etcos⁡t,etsin⁡t,et⟩, 0≤t≤ln⁡4.33
  9. Find r′(t) for r(t)=⟨−3t3+5,sin⁡t,tet⟩r′(t)=⟨−9t2,cos⁡t,(t+1)et⟩
  10. r′(t)=⟨−3t2,2e2t,6t−3⟩ and r(0)=⟨−5,1,0⟩. Find r(t).r(t)=⟨−t3−5,e2t,3t2−3t⟩
  11. Find the unit tangent vector T(1) for r(t)=⟨−t2+8t,−2t2+10t−2,2t2−11t+1⟩.T(1)=⟨611,611,−711⟩
  12. Find the length of r(t)=⟨4cos⁡t,4sin⁡t,3t⟩, 0≤t≤2π.10π
  13. Find r′(t) for r(t)=⟨5t2−4t,sin⁡t,2ln⁡t⟩r′(t)=⟨10t−4,cos⁡t,2t⟩
  14. r′(t)=⟨−6,−6sin⁡2t,9cos⁡3t⟩ and r(0)=⟨0,3,0⟩. Find r(t).r(t)=⟨−6t,3cos⁡2t,3sin⁡3t⟩
  15. Find the unit tangent vector T(1) for r(t)=⟨−2t2+7t−3,t2+2t+1,−2t2+16t+4⟩.T(1)=⟨313,413,1213⟩
  16. Find the length of r(t)=⟨3cos⁡t,3sin⁡t,4t⟩, 0≤t≤π/2.5π2
  17. Find r′(t) for r(t)=⟨−5t2+5t,e−2t,−4t2⟩r′(t)=⟨−10t+5,−2e−2t,−8t⟩
  18. r′(t)=⟨−10t+1,− ⁣sin⁡t,−3t2⟩ and r(0)=⟨0,1,2⟩. Find r(t).r(t)=⟨−5t2+t,cos⁡t,−t3+2⟩
  19. Find the unit tangent vector T(2) for r(t)=⟨2t2−12t+3,2t2−12t+4,−2t2+t+4⟩.T(2)=⟨−49,−49,−79⟩
  20. Find the length of r(t)=⟨2t,t2,ln⁡t⟩, 1≤t≤6.35+ln⁡6

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the unit tangent vector T(1) for r(t)=⟨t,t2,t2−3⟩.

  1. r′(t) = ⟨1, 2t, 2t⟩, so r′(1) = ⟨1, 2, 2⟩.
  2. |r′(1)| = √(1 + 4 + 4) = 3.
  3. T(1) = r′(1)/|r′(1)| = ⟨1/3, 2/3, 2/3⟩.

The answer is ⟨13,23,23⟩.

Where students go wrong

Forgetting the constant vector when integrating. Each component gets its own constant, so the + C is a vector, and an initial condition r(0) fixes all three at once.

Also called vector functions, calculus of vector-valued functions, space curves or derivatives of vector functions.

What you can put on this worksheet

r'(t)
Find r′(t) for r(t)=⟨3t3−5,t3+6⟩ → r′(t)=⟨9t2,3t2⟩
Integrals, and r(t) from its rate
Find ∫⟨9t2,3t2⟩ dt → ⟨3t3,t3⟩+C
Tangent vectors and tangent lines
Find the tangent vector to r(t)=⟨−2t2+16t−2,t2+3t+3⟩ at t=1. → r′(1)=⟨12,5⟩
Arc length of r(t)
Find the length of r(t)=⟨4t+2,4t−3,7t⟩, 1≤t≤2. → 9
Velocity, speed and acceleration
A particle moves with x(t)=2t2+3t and y(t)=−2t3. Find its velocity vector at t=2 → ⟨11,−24⟩

Questions about these worksheets

Yes. Every vector-valued functions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the vector-valued functions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: r'(t), integrals, and r(t) from its rate, tangent vectors and tangent lines, arc length of r(t) and velocity, speed and acceleration. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting the constant vector when integrating. Each component gets its own constant, so the + C is a vector, and an initial condition r(0) fixes all three at once.