1. ## Complex math problems involving calculus, need your help

suppose m(0) milligrams of a drug are put in the blood of an injection. The ammount of drug t hours after the injection is given by
m(t)=m(o)e^-kt, for t > 0, where k is the rate constant, which is related to the half life. we also treat oral administraion
of drugs as an injection, although the model is less accurate because of the drug must be absorbed into the blood through the stomach
4. the sedative diazpam has a half-life of 7 hr. find the drug function m for diazepam.
graph the drug function with m0=5 for 0<t<48 hours.
how much drug remains in the blood 12 hr and 24 hr after a 5-mg dose is taken?
5. the antibiotic tetracycline has a half-life of 9 hours. suppose a doctor wishes a patient to have a 100mg of tetracycline in the blood 18 hours after an injection. what initial does meets his requirement?
6. tweleve hours after a 200 mg dose of a drug is injected. the drug level in the vlood is 75mg. what is the approximate half-life drug?

An intravenous line provides a continuous flow of drug directly into the blood. Assuming no initial drug in the blood, the amount of drug in the blood t hours after the dosing begins
in m(t) - (a/k)(1-e^-kt), for t (=>)0, where k is the rate constant (again related to half life) and A is the rate at which drug flows into the blood (in units of mg/hr)

7.
Suppose an antibiotic with a half life of 12 hours is given to a patient intravenously at a rate of A=50mg/hr.
find the rate constant k
8.
what is the steady-state level of the antibiotic in step 7? that is, evaluate lim(t-> infinity)m(t).
9.
In general, what is the steat state level of a drug delivered by infusion in terms of A and K?
in general at what time does the drug level reach 90% of the steady state level, in terms of A and K?
10.
Based on patients weight, a doctor targets a steady state level of tetracycline of 100mg through infusion.
what infusion rate A should be used? The half life of tetracycline is 9 hr.
11.
in step 10, at what time does the drug level reach 90% of the steady state level? At that time,
how much drug has actually been delivered?
12.
Suppose a patient has been on infusion of tetracycline for 72 hours with infusion rate as found in step 10,
when the delievery is terminated. How long does it take for the drug level in the blood to reach 2mg?

2. ## Re: Complex math problems involving calculus, need your help

that's quite a laundry list of rather basic exponential decay/growth problems.

... what have you done to complete any of them?

3. ## Re: Complex math problems involving calculus, need your help

I got done with the first three parts
1.
.5 = e^1.5k
ln .5 = 1.5 k
ln.5÷1.5 = k
m(4) = 400e^4(ln.5÷1.5)=62.996 mg
m(8) = 400e^8(ln.5÷1.5)=9.921 mg
2.
If half life =t1/2
.5 = e^kt1/2
ln.5 = kt1/2
k=ln.5/t1/2 hr
3.
.01 = e^t(ln.5÷1.5)
ln.01 =t( ln.5 ÷1.5)
ln .01/( ln.5 ÷1.5)=t
9.966 hr=t

4. ## Re: Complex math problems involving calculus, need your help

4 is straightforward ...

$\frac{m_0}{2} = m_0 e^{-7k}$

solve for $k$, store the value in your calculator, graph the function $y = 5e^{-kx}$ for $0 \le x \le 48$, then evaluate the function at $x = 12$ and $x = 24$ as instructed.

5. ## Re: Complex math problems involving calculus, need your help

Okay what about 7-12? Im so lost in those