A dose D of a drug causes a temperature change T in a patient for c a positive constant T is given by T=((c/2)-(D/3)) what is the rate of change of temperature change with respect to dose
For what does the temperature change increase as the dose increase
d T / d D = -1/3
steady if D/3 = c/2
goes up if
c/2 > D/3
or
c = (2/3) D
To find the rate of change of temperature with respect to dose, we need to differentiate the temperature equation with respect to dose, D.
Given:
Temperature, T = (c/2) - (D/3)
To find dT/dD (derivative of temperature with respect to dose):
dT/dD = d/dD ((c/2) - (D/3))
Differentiating each term separately, we get:
dT/dD = 0 - (1/3)
Therefore, the rate of change of temperature with respect to dose is -1/3.
Now, to determine for what values the temperature change increases as the dose increases, we need to find when dT/dD is positive.
Since dT/dD = -1/3, which is negative, it indicates that as the dose increases, the temperature change decreases.
To find the rate of change of temperature change with respect to the dose, we need to differentiate the temperature function T with respect to D.
Given that T = (c/2) - (D/3), let's differentiate both sides with respect to D:
dT/dD = d(c/2)/dD - d(D/3)/dD
The first term, d(c/2)/dD, is equal to 0 because c is a positive constant and does not depend on D. Thus, its derivative with respect to D is zero.
The second term, d(D/3)/dD, is equal to 1/3. This is because D/3 can be written as (1/3)D, which is a constant coefficient times D.
Therefore, the rate of change of temperature change with respect to the dose is 1/3.
Now, to determine when the temperature change increases as the dose increases, we need to analyze the derivative of the temperature function with respect to D.
From above, we found that dT/dD = 1/3, which is a positive constant. This means that the rate of change of temperature change with respect to the dose is always positive.
Therefore, as the dose increases, the temperature change will always increase.