A laboratory scientist monitors the temperature (in ) of a chemical solution during…

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

Problem

A laboratory scientist monitors the temperature

T

(in

^{\circ}\text{C}

) of a chemical solution during the first 15 minutes of a reaction. Analytical equipment provides the following information about the rate of change

T'(t)

of the temperature with respect to time

t

(in minutes): •

T'(t)>0

for

0<t<7

. •

T'(t)=0

t=7

. •

T'(t)<0

for

t>7

. (a) Describe the intervals on which the temperature is increasing and the intervals on which it is decreasing. Clearly reference the given information about

T'(t)

. (b) Determine whether

t=7

minutes corresponds to a relative maximum, a relative minimum, or neither for the temperature. Provide a justification based on the behavior of

T'(t)

on either side of

t=7

. (c) Suppose the temperature at

t=0

22^{\circ}\text{C}

and at

t=15

minutes it is

24^{\circ}\text{C}

. Explain, without additional calculation, whether the maximum temperature attained during the first 15 minutes is greater than

24^{\circ}\text{C}

. Justify your conclusion.

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