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Cogito

Multivariable Calculus · Chapter 2 · Lesson 1

Practice Worksheet

Vector-Valued Functions

One input, three outputs, a curve through space.

12 problems · about 22 minutes · N-VM.A.1, F-IF.B.4

NameDate

Figure — use these to answer the problems

The point at angle t has coordinates ⟨cos t, sin t⟩. That is the first two components of a helix.
(0.500, 0.866)
  • Angle60° = π/3 rad
  • x-coordinate0.500
  • y-coordinate0.866
  • cos 60°0.500
  • sin 60°0.866
the parameter t, in degrees60
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. r(t) = ⟨t², 2t, 0⟩ at t = 4. What is the y-coordinate?

    Answer

  2. What does the parameter t usually represent?

    • Time, so r(t) is where the particle is at that moment.
    • The distance from the origin.

  3. r(t) = ⟨3t, t², 1⟩ at t = 2. What is the x-coordinate?

    Answer

  4. r(t) = ⟨3t, t², 1⟩ at t = 2. What is the y-coordinate?

    Answer

  5. r(t) = ⟨cos t, sin t, 0⟩. What is the distance from the origin at any t?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. How many full circles does ⟨cos t, sin t, 0⟩ trace as t runs from 0 to 4π?

    Answer

  2. r(t) = ⟨t, t, t⟩. Is this a straight line? 1 yes, 0 no.

    Answer

  3. r(t) = ⟨cos t, sin t, 4t⟩. How much does z rise over one full turn, using 2π as about 6.28?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Order the values of t so the particle on ⟨t, t², 0⟩ visits them in sequence.

    Write 1 to 4 in the boxes to put these in order.

    • t = −1
    • t = 0
    • t = 1
    • t = 2

  2. r(t) = ⟨5, t, 0⟩. What is the x-coordinate at t = 100?

    Answer

  3. The Spring

    r(t) = ⟨cos t, sin t, 2t⟩. What is the z-coordinate at t = 3?

    Answer

  4. The Start

    r(t) = ⟨t + 1, 3t, t³⟩. What is the x-coordinate at t = 0?

    Answer