A dose, D, of a drug causes a temperature change, T, in apatient. For C a positive constant, t is given by T=((C/2)-(D/3))D^3.
(a) What is the rate of change of temperature change with respect to dose.
(b) For what dose does the temperature change increase as the dose increases.
For part a I got CD^2-D^3=0
For part b I got c^2/4
Any help would be greatly appreciated.
dT/dD= (C/2-D/3)3D^2 + D^3(-1/3)
= 3/2 CD^2 + D^3(2/3)
check that.
for the second part,
set dT/dD to zero, and solve for D.
To find the rate of change of temperature, we need to take the derivative of the temperature equation with respect to the dose, which is D.
(a) Rate of change of temperature with respect to dose:
To find the derivative of T with respect to D, we use the power rule of differentiation.
The power rule states that if we have a function of the form f(D) = aD^n, where a and n are constants, the derivative is f'(D) = anD^(n-1).
In this case, we have T = ((C/2) - (D/3))D^3.
To find the derivative of T with respect to D, we apply the power rule:
T' = dT/dD = [(C/2) - (D/3)] * 3D^2 + D^3 * [-1/3]
= (3CD^2/2) - (9D^3/6) - (D^3/3)
= (3CD^2/2) - (3D^3/2) - (D^3/3)
Therefore, the rate of change of temperature with respect to dose is T' = (3CD^2/2) - (3D^3/2) - (D^3/3).
(b) To find the dose at which the temperature change increases as the dose increases, we need to find the interval where the rate of change of temperature is greater than zero (T' > 0).
Setting T' > 0:
(3CD^2/2) - (3D^3/2) - (D^3/3) > 0
Combine the like terms to get:
(3CD^2 - 3D^3 - 2D^3)/2 > 0
(3CD^2 - 5D^3)/2 > 0
Since we want the temperature change to increase, we need the expression to be positive.
For (3CD^2 - 5D^3)/2 to be positive, both the numerator and denominator must have the same sign or be zero.
Case 1: Numerator and denominator are both positive or zero:
3CD^2 - 5D^3 >= 0
D(3C - 5D^2) >= 0
In this case, either D and (3C - 5D^2) are both positive or zero, or both negative or zero.
Case 2: Numerator and denominator are both negative or zero:
3CD^2 - 5D^3 <= 0
D(3C - 5D^2) <= 0
In this case, either D and (3C - 5D^2) are both negative or zero, or both positive or zero.
To summarize, for the temperature change to increase as the dose increases, either both D and (3C - 5D^2) must be positive or zero, or both negative or zero.