A differentiable function models the position (in meters) of a bicycle along a track
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A differentiable function s models the position (in meters) of a bicycle along a track, where t is measured in seconds. The graph of s versus t is described verbally as follows:
• From t=0 to t=2, the graph rises steeply and concavely upward.
• Between t=2 and t=4, the graph continues to rise but at a decreasing rate, becoming nearly flat at t=4.
• After t=4, the graph descends gently.
Consider the secant line that joins the points (1,s(1)) and (4,s(4)). The slope of this secant line is 2.
(a) Interpret the numerical value 2 in the context of the bicycle’s motion. Include appropriate units.
(b) Without computing any derivatives, determine whether the instantaneous velocity of the bicycle at t=3 seconds is greater than, less than, or equal to 2. Clearly explain your reasoning based on the given description of the graph.
(c) Rank the instantaneous velocities at t=1, t=3, and t=4 from greatest to least. Justify your ranking using features of the graph such as steepness and concavity.