Let us revisit a question from a few problem sets ago with slightly different numbers. A patient takes 200 mg of an antibiotic every 6 hours. The half-life of the drug (the time it takes for half of the drug to be eliminated from the blood) is 6 hours. Let dn denote the amount (in mg) of medication in the bloodstream after n doses, where d₁ = 200. 1. Explain why dn+1 = 0.5dn + 200, d₁ = 200 is the correct recurrence relation. 2. Use the recurrence relation to write down the expressions for d2, d3, and d4. Do not simplify- instead, keep powers of 0.5 in your answers. 3. You should see that each dn is a geometric sum. We are interested in finding the long-term (steady state) amount of antibiotic in your blood, i.e., limno dn. Write this limit as an infinite series using summation (E) notation. 4. Evaluate the geometric series from the previous part using the geometric series formula. 5. Previously we used the recurrence relation dn+1 = 0.5dn + 200, d₁= 200 and the assumption that dnL to compute the limiting value. Compute L is this way and compare to your answer in 4.
Let us revisit a question from a few problem sets ago with slightly different numbers. A patient takes 200 mg of an antibiotic every 6 hours. The half-life of the drug (the time it takes for half of the drug to be eliminated from the blood) is 6 hours. Let dn denote the amount (in mg) of medication in the bloodstream after n doses, where d₁ = 200. 1. Explain why dn+1 = 0.5dn + 200, d₁ = 200 is the correct recurrence relation. 2. Use the recurrence relation to write down the expressions for d2, d3, and d4. Do not simplify- instead, keep powers of 0.5 in your answers. 3. You should see that each dn is a geometric sum. We are interested in finding the long-term (steady state) amount of antibiotic in your blood, i.e., limno dn. Write this limit as an infinite series using summation (E) notation. 4. Evaluate the geometric series from the previous part using the geometric series formula. 5. Previously we used the recurrence relation dn+1 = 0.5dn + 200, d₁= 200 and the assumption that dnL to compute the limiting value. Compute L is this way and compare to your answer in 4.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Let us revisit a question from a few problem sets ago with slightly different numbers. A patient
takes 200 mg of an antibiotic every 6 hours. The half-life of the drug (the time it takes for half
of the drug to be eliminated from the blood) is 6 hours. Let dn denote the amount (in mg) of
medication in the bloodstream after n doses, where d₁= 200.
1. Explain why dn+1 = 0.5dn + 200, d₁= 200 is the correct recurrence relation.
2. Use the recurrence relation to write down the expressions for d2, d3, and d4. Do not simplify-
instead, keep powers of 0.5 in your answers.
3. You should see that each dn is a geometric sum. We are interested in finding the long-term
(steady state) amount of antibiotic in your blood, i.e., limno dn. Write this limit as an
infinite series using summation (E) notation.
4. Evaluate the geometric series from the previous part using the geometric series formula.
5. Previously we used the recurrence relation dn+1 = 0.5dn + 200, d₁ = 200 and the assumption
that dn→ L to compute the limiting value. Compute L is this way and compare to your
answer in 4.
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