Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus cavity while simultaneously being filtered out of the blood by the liver. The following is a model for the concentration (in µg/mL) of the antibiotic in the sinus cavity as a function of time (in hours) since the injection. C(t) = (e−αt − e −βt )/(β − α) , where α and β are constants with β > α > 0. (a) Using the first derivative test, find when the maximum concentration occurs. (Your argument must use the first derivative test.) (b) When does the rate of change of concentration begin to increase? (Your answer should be supported by a rigorous argument.)
Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus cavity while simultaneously being filtered out of the blood by the liver. The following is a model for the concentration (in µg/mL) of the antibiotic in the sinus cavity as a function of time (in hours) since the injection.
C(t) = (e−αt − e −βt )/(β − α) ,
where α and β are constants with β > α > 0.
(a) Using the first derivative test, find when the maximum concentration occurs. (Your argument must use the first derivative test.)
(b) When does the rate of change of concentration begin to increase? (Your answer should be supported by a rigorous argument.)
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