MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A skateboarder moves along a straight, horizontal stretch of pavement. A motion-sensor records the skateboarder’s acceleration from t=0 to t=10 seconds. The acceleration graph is described verbally as follows:
• From t=0 to t=4 seconds, acceleration is constant at 2 m/s2.
• From t=4 to t=7 seconds, acceleration decreases linearly to −1 m/s2.
• From t=7 to t=10 seconds, acceleration remains constant at −1 m/s2.
At t=0, the skateboarder’s velocity is 1 m/s and her position is 0 m.
(a) Without performing any calculations, describe qualitatively how the skateboarder’s speed changes over the time interval 0≤t≤10. Justify your description by referring to features of the acceleration graph.
(b) Compute the exact change in velocity over each sub-interval defined above, and determine the skateboarder’s velocity at t=4, t=7, and t=10 seconds.
(c) Determine the skateboarder’s position at t=4, t=7, and t=10 seconds.
(d) During which time interval is the skateboarder farthest from the starting point? Justify your answer using both position values and the sign of velocity.