AP Calculus AB:: Contextual Applications

If \(s(t)\) gives the position of an object at time \(t\), what does \(s'(t)\) represent?

- **A)** The object's total distance traveled
- **B)** The object's acceleration at time \(t\)
- **C)** The object's velocity at time \(t\)
- **D)** The object's position at time \(t\)

Given velocity \(v(t)\), what does \(v'(t)\) represent?

- **A)** Acceleration, \(a(t) = v'(t) = s''(t)\)
- **B)** Speed
- **C)** Displacement
- **D)** Position

Given \(s(t) = t^3 - 6t^2 + 9t\), find the velocity function \(v(t)\).

- **A)** \(v(t) = t^2 - 6t + 9\)
- **B)** \(v(t) = 3t^2 - 12t\)
- **C)** \(v(t) = 3t^2 - 12t + 9\)
- **D)** \(v(t) = 3t^2 - 6t\)

Using \(v(t) = 3t^2 - 12t + 9\) (from \(s(t)=t^3-6t^2+9t\)), find the acceleration function \(a(t)\).

- **A)** \(a(t) = 6t + 12\)
- **B)** \(a(t) = 6t - 12\)
- **C)** \(a(t) = 6t\)
- **D)** \(a(t) = 3t - 12\)

For \(v(t) = 3t^2 - 12t + 9\), at what time(s) \(t \ge 0\) is the object at rest (\(v(t)=0\))?

- **A)** \(t=2\) only
- **B)** \(t=0\) only
- **C)** \(t=1\) and \(t=3\)
- **D)** \(t=1\) only

An object's velocity changes sign from positive to negative at \(t=1\). What does this tell you about the object's motion?

- **A)** The object speeds up at \(t=1\)
- **B)** The object is accelerating at a constant rate
- **C)** The object reverses direction at \(t=1\), momentarily stopping there before moving the opposite way
- **D)** The object's position is zero at \(t=1\)

What is the relationship between speed and velocity?

- **A)** Speed \(= -v(t)\) always
- **B)** Speed \(= |v(t)|\), the absolute value of velocity, and is always non-negative
- **C)** Speed is the derivative of velocity
- **D)** Speed and velocity are always identical, including sign

An object has \(v(t) = -4\) and \(a(t) = -2\) at a given instant (same sign). What does this indicate about the object's speed at that instant?

- **A)** The object is momentarily at rest
- **B)** The object is changing direction
- **C)** The object is speeding up, since velocity and acceleration share the same sign
- **D)** The object is slowing down