Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′ ( t ) + km ( t ) = I , where m ( t ) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. a. Show by substitution that if the initial mass of drug in the blood is zero ( m (0) = 0), then the solution of the initial value problem is m ( t ) = I k ( 1 − e − k t ) . b. Graph the solution for I = 10 mg/hr and k = 0.05 hr –1 . c. Evaluate lim t → ∞ m ( t ) , the steady-state drug level, and verify the result using the graph in part (b).
Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′ ( t ) + km ( t ) = I , where m ( t ) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. a. Show by substitution that if the initial mass of drug in the blood is zero ( m (0) = 0), then the solution of the initial value problem is m ( t ) = I k ( 1 − e − k t ) . b. Graph the solution for I = 10 mg/hr and k = 0.05 hr –1 . c. Evaluate lim t → ∞ m ( t ) , the steady-state drug level, and verify the result using the graph in part (b).
Solution Summary: The author explains that if the initial mass of the drug in the blood is zero, then satisfies the differential equation.
Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t) + km(t) = I, where m(t) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate.
a. Show by substitution that if the initial mass of drug in the blood is zero (m(0) = 0), then the solution of the initial value problem is
m
(
t
)
=
I
k
(
1
−
e
−
k
t
)
.
b. Graph the solution for I = 10 mg/hr and k = 0.05 hr–1.
c. Evaluate
lim
t
→
∞
m
(
t
)
, the steady-state drug level, and verify the result using the graph in part (b).
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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