Question

for some positive constant c, a patients temperature change, t, due to a dose, d, of a drug is given by t = (\frac{c}{2}-\frac{d}{3})d^{2}. what dosage maximizes the temperature change? d = the sensitivity of the body to the drug is defined as dt/dd. what dosage maximizes sensitivity? d =

Explanation:

Step1: Expand the function

Expand \(T = (\frac{C}{2}-\frac{D}{3})D^{2}\) to \(T=\frac{C}{2}D^{2}-\frac{1}{3}D^{3}\).

Step2: Find the first - derivative for temperature change

Differentiate \(T\) with respect to \(D\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(T^\prime=\frac{dT}{dD}=C D - D^{2}\).
Set \(T^\prime = 0\), so \(CD - D^{2}=D(C - D)=0\).
Solving \(D(C - D)=0\) gives \(D = 0\) or \(D = C\).
Differentiate \(T^\prime\) to get \(T^{\prime\prime}=C-2D\).
When \(D = 0\), \(T^{\prime\prime}=C>0\) (minima). When \(D = C\), \(T^{\prime\prime}=C - 2C=-C<0\) (maxima).

Step3: Find the function for sensitivity

Since sensitivity \(S=\frac{dT}{dD}=CD - D^{2}\).
Differentiate \(S\) with respect to \(D\): \(S^\prime=C - 2D\).
Set \(S^\prime=0\), then \(C - 2D = 0\).
Solving for \(D\) gives \(D=\frac{C}{2}\).
Differentiate \(S^\prime\) to get \(S^{\prime\prime}=-2<0\).

Answer:

For maximizing temperature change: \(D = C\).
For maximizing sensitivity: \(D=\frac{C}{2}\).