A medicine is administered to a patient. The amount of medicine M, in milligrams, in 1 mL of the patient’s blood, t hours after the injection, is
M(t)=-1/3t^2+t where 0<t<3
a. Find the rate of change in the amount M, 2h after the injection.
b. What is the significance of the fact that your answer is negative?
a)
Dependent: amount of medicine, mg
Independent: time, hour
f(x2) ---> -1/3(2)^2+2=2/3, f(x1)---> -1/3(0)^2+0=0
slope of a secant= f(x2)-f(x1) over x2 - x1
2/3 - 0 over 2 - 0 = 1/3mg/h
b) amount of medicine in 1 mL of blood being dissipated throughout the system
-(1/3) t^2 + t
or
-1/(3 t^2) + t
or
1/(3t^2+t)
To find the rate of change in the amount M, 2 hours after the injection, we need to take the derivative of the given function M(t) with respect to t.
a. Find the rate of change in the amount M, 2 hours after the injection:
Step 1: Take the derivative of M(t) with respect to t:
M'(t) = d(M(t))/dt = d/dt(-1/3t^2 + t)
Step 2: Compute the derivative using the power rule of differentiation:
M'(t) = -2/3t + 1
Step 3: Substitute t = 2 (since we are interested in 2 hours after the injection) into the derivative:
M'(2) = -2/3(2) + 1
M'(2) = -4/3 + 1
M'(2) = -1/3
Therefore, the rate of change in the amount M, 2 hours after the injection, is -1/3 milligrams per hour.
b. The significance of the fact that our answer is negative means that the amount of medicine M is decreasing over time. In this case, the rate of decrease is 1/3 milligrams per hour. It indicates that the medicine is being metabolized or eliminated by the body, resulting in a decrease in its concentration in the patient's blood.
If the first then
dM/dt = -(1/3)(2t) + 1
= -2t/3 + 1
if t = 2
dM/dt = -4/3 + 3/3 = -1/3
the medicine is disappearing, presumably because it is eating bad guys.