Cogito
Multivariable Calculus · Chapter 2 · Lesson 1
Vector-Valued Functions
One input, three outputs, a curve through space.
12 problems · about 22 minutes · N-VM.A.1, F-IF.B.4
What this lesson teaches
The student describes and evaluates a curve given by a vector-valued function.
- A vector-valued function turns one number into a point, tracing a curve.
- ⟨cos t, sin t, 0⟩ is a circle; adding a rising z makes it a helix.
- The same path can be traced by different parametrisations at different speeds.
Warm Up
Straightforward practice. Get the method working first.
5 problemsr(t) = ⟨t², 2t, 0⟩ at t = 4. What is the y-coordinate?
Answer 8
Why 8.
What does the parameter t usually represent?
Answer Time, so r(t) is where the particle is at that moment.
Why It is a clock reading.
r(t) = ⟨3t, t², 1⟩ at t = 2. What is the x-coordinate?
Answer 6
Why 3 × 2.
r(t) = ⟨3t, t², 1⟩ at t = 2. What is the y-coordinate?
Answer 4
Why 2 squared.
r(t) = ⟨cos t, sin t, 0⟩. What is the distance from the origin at any t?
Answer 1
Why cos² + sin² = 1.
Build It Up
The same ideas with more to keep track of.
3 problemsHow many full circles does ⟨cos t, sin t, 0⟩ trace as t runs from 0 to 4π?
Answer 2
Why One circle takes 2π.
r(t) = ⟨t, t, t⟩. Is this a straight line? 1 yes, 0 no.
Answer 1
Why All three components grow together.
r(t) = ⟨cos t, sin t, 4t⟩. How much does z rise over one full turn, using 2π as about 6.28?
Answer 25.12
Why 4 times 6.28.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder the values of t so the particle on ⟨t, t², 0⟩ visits them in sequence.
Answer 1. t = −1 2. t = 0 3. t = 1 4. t = 2
Why The parameter runs from small to large.
r(t) = ⟨5, t, 0⟩. What is the x-coordinate at t = 100?
Answer 5
Why A constant component never moves.
The Spring: r(t) = ⟨cos t, sin t, 2t⟩. What is the z-coordinate at t = 3?
Answer 6
Why 2 × 3 = 6.
The Start: r(t) = ⟨t + 1, 3t, t³⟩. What is the x-coordinate at t = 0?
Answer 1
Why 1.