A curve representing the depth (in meters) of a submarine below sea level as a…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A curve representing the depth d=f(t) (in meters) of a submarine below sea level as a function of time t (in minutes) is displayed on a monitor. At t=6 min, three straight lines are overlaid on the screen, each intersecting the curve at the point (6,50).
Line A passes through (8,46), Line B passes through (9,37), and Line C passes through (7,57). Only one of these lines is tangent to the curve at t=6.
Complete the tasks below.
(a) Determine which line (A, B, or C) is the best candidate to be the tangent line at (6,50). Support your choice by computing each line’s slope and discussing which is most consistent with the submarine’s nearby motion as described below:
• From t=4 to t=5, the submarine rose steadily with an average slope of about −6 m/min.
• From t=7 to t=8, it continued rising but more slowly, with an average slope of about −4 m/min.
(b) Using the line you selected in part (a), estimate f′(6). Show your calculation, including units.
(c) Explain how the instantaneous rate of ascent you found in part (b) compares to the average rates given for adjacent intervals, and discuss what this implies about the submarine’s change in vertical speed around t=6.