The altitude (in meters) of a balloon above the ground is a differentiable function…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The altitude A (in meters) of a balloon above the ground is a differentiable function of time t (in minutes). The balloon’s vertical velocity is given by A′(t), whose graph is described below for 0≤t≤18.
• A′(t)=0 at t=3,8,14,18 and only at these values.
• On (0,3), A′(t)>0 and is decreasing.
• On (3,6), A′(t)<0 and is decreasing.
• On (6,8), A′(t)<0 and is increasing.
• On (8,11), A′(t)>0 and is increasing.
• On (11,14), A′(t)>0 and is decreasing.
• On (14,18), A′(t)<0 and is decreasing.
Assume A(0)=120 m.
(1) Determine the time intervals during which the balloon is rising and the time intervals during which it is descending. Express your answers in interval notation and justify each interval by referencing the sign of A′(t). No calculation of altitude is required.
(2) Identify the times at which the balloon reaches a local maximum altitude and the times at which it reaches a local minimum altitude on the interval 0<t<18. Classify each critical time using an appropriate test based on A′(t) and provide supporting reasoning.
(3) Using your conclusions from parts (a) and (b), construct a rough sketch of a possible graph of A(t) for 0≤t≤18. Your sketch must respect the initial condition and the relative extrema you identified. Label each extremum and indicate approximate changes in altitude where appropriate.