The height (in meters) of a hot-air balloon above the ground is recorded every few…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The height h (in meters) of a hot-air balloon above the ground is recorded every few tenths of a minute during a smooth ascent. A portion of the data is shown.
t (min) | h (m)
7.5 | 660
7.9 | 668
8.0 | 670
8.1 | 671
8.5 | 674
Assume the balloon’s altitude is a differentiable function of time on the interval [7.5,8.5].
(a) Compute the balloon’s average vertical speed (in m/min) on each of the following intervals: [7.5,8.5], [7.9,8.1], and [8.0,8.1]. Show your calculations.
(b) Using successive shrinkage of the time interval about t=8, refine your estimate of the balloon’s instantaneous vertical speed at t=8 minutes. State your final estimate to the nearest 0.1 m/min.
(c) Interpret the meaning of the value obtained in part (b) in the context of the balloon’s motion. Include units in your answer and comment on whether the balloon is ascending faster or slower than it was earlier in the listed data.