Mathematics

Problem

A pharmaceutical tablet releases medication according to the schedule

D(t)=\begin{cases} 0, & t<0,\\ 50, & 0\le t<4,\\ kt+b, & t\ge 4, \end{cases}

where

t

is time in hours and

D(t)

is dosage rate in milligrams per hour. The constants

k

and

b

may be adjusted by the manufacturer. (a) Suppose

k=0

and

b=75

. Determine

\displaystyle \lim_{t\to 4^-}D(t)

and

\displaystyle \lim_{t\to 4^+}D(t)

, then calculate the jump size at

t=4

. (b) Find all pairs

(k,b)

for which

D

is continuous at

t=4

. Fully justify your answer. (c) Using the parameters found in part (b), describe in words how the medication delivery rate changes over time and why the jump has been eliminated.

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