Cogito
AP Calculus BC · Chapter 1 · Lesson 3
Vector-Valued Functions
Position, velocity and acceleration as vectors.
12 problems · about 22 minutes · CHA-3.G
What this lesson teaches
The student differentiates and integrates vector-valued functions in a motion context.
- Differentiate a vector-valued function componentwise.
- Speed is the magnitude of velocity and is never negative.
- Integrating speed gives distance; integrating velocity gives displacement.
Warm Up
Straightforward practice. Get the method working first.
5 problemsVelocity components 9 and 12. What is the speed?
Answer 15
Why 15.
What is the difference between distance travelled and displacement?
Answer Distance integrates speed; displacement integrates velocity and can be zero.
Why One has no sign; the other does.
Velocity components 6 and 8. Speed?
Answer 10
Why √100.
Velocity components 5 and 12. Speed?
Answer 13
Why √169.
Position (t², t³) gives velocity (2t, 3t²). y component at t = 2?
Answer 12
Why 3 × 4.
Build It Up
The same ideas with more to keep track of.
3 problemsCan speed be negative? 1 yes, 0 no.
Answer 0
Why It is a magnitude.
Integrating speed gives what? 1 distance travelled, 2 displacement.
Answer 1
Why Speed has no sign.
A particle returns to its start. What is its displacement magnitude?
Answer 0
Why Start and end coincide.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each quantity by whether it is a vector.
Answer Vector: Velocity, Acceleration · Scalar: Speed, Distance travelled
Why Two of these carry a direction.
Velocity components 8 and 15. Speed?
Answer 17
Why √289.
The Speed: Velocity components 3 and 4. What is the speed?
Answer 5
Why 5.
The Velocity: Position (t², t³) gives velocity (2t, 3t²). What is the x component at t = 3?
Answer 6
Why 6.