The position (in meters) of a skateboarder moving along a straight ramp is recorded…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The position s (in meters) of a skateboarder moving along a straight ramp is recorded every 0.25 seconds, producing a smooth position–time graph that can be approximated by the following verbal description:
• From t=0 s to t=1 s the graph is concave up, starting at s=0 m and reaching s=4 m; the tangent slope increases steadily from 2 m/s to 6 m/s.
• From t=1 s to t=2.5 s the graph is concave down, still increasing but with the tangent slope decreasing from 6 m/s to 0 m/s at t=2.5 s.
• From t=2.5 s to t=3 s the graph turns downward, remaining concave down and decreasing to s=2 m at t=3 s. Over this interval the tangent slope decreases from 0 m/s to −4 m/s.
Assume that between the listed data points the graph is twice differentiable and follows the qualitative shape described.
(a) Sketch a qualitative graph of the velocity function v(t)=s′(t) on the interval 0≤t≤3. Clearly label the values of v at t=0,1,2.5, and 3 seconds.
(b) On the same set of axes, sketch a qualitative graph of the acceleration function a(t)=v′(t) and indicate the sign of a(t) on each subinterval given in the description.
(c) Identify all times in (0,3) at which a(t)=0 and classify each as a possible local maximum, local minimum, or point of inflection for the velocity function. Justify your classifications using changes in concavity and reasoning about slope.
(d) Discuss the physical meaning of the sign of a(t) for the skateboarder’s motion, specifically relating positive versus negative acceleration to the rider’s experience on the ramp.