A pharmaceutical tablet releases medication according to the schedule D(t)= cases 0,…

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A pharmaceutical tablet releases medication according to the schedule D(t)={0,t<0,50,0≤t<4,kt+b,t≥4,D(t)=\begin{cases} 0, & t<0,\\ 50, & 0\le t<4,\\ kt+b, & t\ge 4, \end{cases} where tt is time in hours and D(t)D(t) is dosage rate in milligrams per hour. The constants kk and bb may be adjusted by the manufacturer. (a) Suppose k=0k=0 and b=75b=75. Determine lim⁡t→4−D(t)\displaystyle \lim_{t\to 4^-}D(t) and lim⁡t→4+D(t)\displaystyle \lim_{t\to 4^+}D(t), then calculate the jump size at t=4t=4. (b) Find all pairs (k,b)(k,b) for which DD is continuous at t=4t=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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