Cogito
AP Calculus AB · Chapter 4 · Lesson 1
Motion and Related Rates
Derivatives with units attached.
11 problems · about 25 minutes · AP Calculus AB 4.1, 4.4, 4.5
What this lesson teaches
The student interprets derivatives in motion contexts and sets up a related-rates problem.
- Position differentiates to velocity, and velocity to acceleration.
- Speed is the magnitude of velocity and is never negative.
- An object speeds up when velocity and acceleration share a sign.
Warm Up
Straightforward practice. Get the method working first.
4 problemsVelocity −4, acceleration 6. Speeding up? 1 for yes, 0 for no.
Answer 0
Why Slowing down.
Velocity 5, acceleration 2. Speeding up? 1 for yes, 0 for no.
Answer 1
Why Same sign.
Velocity 5, acceleration −2. Speeding up? 1 for yes, 0 for no.
Answer 0
Why Opposite signs.
Velocity −6, acceleration −3. Speeding up? 1 for yes, 0 for no.
Answer 1
Why Same sign.
Build It Up
The same ideas with more to keep track of.
3 problemsOrder the steps of a related-rates problem.
Answer 1. Write a relationship between the quantities 2. Differentiate both sides with respect to time 3. Substitute the known values and rates 4. Solve for the unknown rate
Why Substituting numbers before differentiating destroys the relationship.
Position s(t) = t². What is the velocity at t = 3?
Answer 6
Why 2t.
Same position function. What is the acceleration?
Answer 2
Why Differentiate 2t.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsVelocity −3. Is the object moving forwards? 1 for yes, 0 for no.
Answer 0
Why The sign is direction.
Speed is 9 and the object moves backwards. What is the velocity?
Answer -9
Why Attach the direction.
Speed: Velocity is −7. What is the speed?
Answer 7
Why 7.
At Rest: An object is momentarily at rest. What is its velocity?
Answer 0
Why 0.