A high-altitude balloon’s height (in meters) is recorded every second

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

A high-altitude balloon’s height

h

(in meters) is recorded every second. Selected data are shown below.

Time t (s)	13	14	15	16	17
h(t) (m)	48.7	52.1	55.6	59.0	62.3

For

13\le t\le 17

, the balloon’s vertical acceleration satisfies

|h''(t)|\le 0.50\text{ m/s}^2

. Use the data to analyze the balloon’s instantaneous vertical velocity at

t=15\text{ s}

. (a) Estimate

h'(15)

using the slope of the secant line through the points at

t=14\text{ s}

and

t=16\text{ s}

. Include appropriate units. (b) Estimate

h'(15)

using the slope of the secant line through the points at

t=13\text{ s}

and

t=17\text{ s}

. Include appropriate units. (c) Decide which estimate in parts (a) and (b) is expected to be closer to

h'(15)

. Justify your answer, making explicit reference to the bound on

h''(t)

. (d) Using the estimate you selected in part (c) and the bound on

h''(t)

, determine a numerical interval that must contain the true value of

h'(15)

. Show all reasoning, including any theorems invoked.