A company tracks the temperature (in degrees Celsius) of a chemical solution over…

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

Problem

A company tracks the temperature

T

(in degrees Celsius) of a chemical solution over time

t

(in minutes). Between

t=0

and

t=14

, the recorded values of

T

lie on three consecutive straight-line segments that join the points

(0,20)

(6,20)

(10,35)

, and

(14,35)

. Assume all intermediate points on each straight segment are included in the graph. Throughout this question, interpret

\dfrac{dT}{dt}

as the instantaneous rate of change of temperature. (a) Determine

\dfrac{dT}{dt}

for all

t

in the open interval

(0,6)

. Include units in your answer. (b) Determine

\dfrac{dT}{dt}

for all

t

in the open interval

(6,10)

. (c) Does

\dfrac{dT}{dt}

exist at

t=6

? Explain your reasoning without using calculus notation other than the limit definition of derivative. (d) Suppose a safety guideline limits the instantaneous heating rate to no more than

3

°C per minute. Identify every sub-interval of

(0,14)

on which the process violates this guideline, and justify your answer.

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