A temperature probe moves along a 12-meter steel bar according to the position function

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

x(t)

, where

x

is the distance in meters from the left end of the bar at time

t

seconds. The probe’s velocity is given by

\frac{dx}{dt}=0.04t^{2}-0.30t+1.20.

The temperature distribution along the bar is described by the differentiable function

T(x)=\frac{120}{1+e^{-0.6(x-6)}},

where

T

is measured in °C and

0\le x\le12

. Assume

x(0)=1.5

m and that the probe never leaves the bar during the interval

0\le t\le8

. (a) Write an explicit expression for

\dfrac{dT}{dt}

in terms of

t

by differentiating the composite function

T(x(t))

. Do not substitute a particular value of

t

yet. (b) Compute

\dfrac{dT}{dt}

t=4

seconds. Include units. (c) Using your result from part (a), determine all times in the open interval

0<t<8

for which the probe’s temperature reading is increasing at the fastest rate. Justify your answer, clearly stating any critical points and how you classified them. (d) Describe physically what it means for the probe to experience the greatest positive value of

\dfrac{dT}{dt}

in this context.

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