To determine
a)To Show: That limt→∞C(t)=0,where C(t)=K(e−at−e−bt)
Answer
The value of the limit is limt→∞C(t)=0
Explanation
Explanation: We know that as x→∞,e−∞→0
Therefore,
limt→∞K(e−at−e−bt)=Klimt→∞(e−at−e−bt)=K(0−0)=0
Final statement: limt→∞C(t)=0
To determine
b)To find: C′(t), the rate of change of drug concentration in the blood
Explanation
Given C(t)=K(e−at−e−bt)
Differentiating with respect to t, we get
C′(t)=K(−ae−at−(−be−bt))
C′(t)=K(−ae−at+be−bt)
Final statement: C′(t)=K(−ae−at+be−bt)
To determine
c) To find:When is the rate equal to zero.
Answer
When the rate is equal to zero, we get t=lna−lnba−b
Explanation
Given C(t)=K(e−at−e−bt)
Differentiating with respect to t, we get
C′(t)=K(−ae−at−(−be−bt))
C′(t)=K(−ae−at+be−bt)
C′(t)=K(−ae−at+be−bt)=0⇒−ae−at+be−bt=0⇒be−bt=ae−at⇒e−bte−at=ab⇒eat.e−bt=ab⇒e(a−b)t=ab⇒(a−b)t=ln(ab)⇒t=lna−lnba−b
Final statement: t=lna−lnba−b