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