During the first 10 seconds of a roller-coaster ride
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
During the first 10 seconds of a roller-coaster ride, the vertical position of the car, in meters, is modeled by a continuous function s whose graph is described below.
• The car starts at ground level: s(0)=0.
• From t=0 to t=4 s the car climbs the initial hill, rising concave down to reach a height of 30 m at t=4 s where it momentarily levels off.
• Between t=4 s and t=7 s the car descends into a valley, reaching its lowest point of 5 m at t=7 s. The graph is concave up on this interval.
• From t=7 s to t=10 s the car climbs again, reaching 15 m at t=10 s; the slope of the graph is positive but decreasing on this final climb.
Let v(t)=s′(t) denote the vertical velocity of the car (positive upward).
(a) Using the description above, determine the sign of v(t) on each of the intervals (0,4), (4,7), and (7,10). Explain how the motion of the car leads to each conclusion.
(b) Identify all times in 0≤t≤10 at which v(t)=0 or v(t) is undefined. For each time, state the physical significance regarding the car’s motion.
(c) Sketch a qualitative graph of y=v(t) on the interval [0,10]. Your sketch should clearly indicate zeros, relative extrema, and any discontinuities, and it should be consistent with the concavity information for s.
(d) On your sketch from part (c), indicate the intervals where the acceleration a(t)=v′(t) is positive and where it is negative. Justify your conclusions using features of v(t).
(e) Briefly explain how the shapes of the graphs of s and v together illustrate the relationship between a function, its first derivative, and its second derivative in this real-world context.