A reliable method
Differentiate position with respect to time. To find total distance, split the interval wherever velocity changes sign and add absolute displacements.
Worked foundation example
An object has position s(t)=2t²+4t meters. Find its velocity at t=3 seconds. Velocity is s′(t)=4t+4. Evaluate at t=3. The velocity is 16 meters per second.
The common trap
A negative velocity does not by itself mean an object is slowing down. Compare the signs of velocity and acceleration.
Connect the representations
Displacement is signed final position minus initial position. Total distance is nonnegative and cannot be smaller than the magnitude of displacement.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare displacement with distance
For velocity v(t)=2t−4 on [0,4], the object reverses direction at t=2. The integral on [0,2] is −4 and the integral on [2,4] is 4, so displacement is zero. Total distance is |−4|+|4|=8. A final position equal to the initial position does not imply the object stayed still. The sign of velocity describes direction, while speed is its magnitude.
Transfer and explain
Transfer: change the interval to [0,3]. The first signed area is −4 and the second is 1, so displacement is −3 and distance is 5. Identify the splitting time before integrating.
An object has position s(t)=2t²+4t meters. Find its velocity at t=3 seconds.
- Velocity is s′(t)=4t+4.
- Evaluate at t=3.
- The velocity is 16 meters per second.
Compare the other answer choices
Report acceleration instead of velocity.
Drop the derivative of the linear term.
Report position instead of velocity.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus