Sequences have many applications in mathematics and the sciences. In a recent paper³ the authors write The incretin hormone glucagon-like peptide-1 (GLP-1) is capable of amelio- rating glucose-dependent insulin secretion in subjects with diabetes. How- ever, ‚ its very short half-life (1.5-5 min) in plasma represents a major limita- tion for its use in the clinical setting. The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium. For this exercise, assume the half-life of GLP-1 is 5 minutes. So if A is the amount of GLP-1 in plasma at some time t, then only 4 of the hormone will be present after t + 5 minutes. Suppose A₁ = 100 grams of the hormone are initially present in plasma. a. Let A₁ be the amount of GLP-1 present after 5 minutes. Find the value of A₁. b. Let A₂ be the amount of GLP-1 present after 10 minutes. Find the value of A₂. c. Let A3 be the amount of GLP-1 present after 15 minutes. Find the value of A3. d. Let A be the amount of GLP-1 present after 20 minutes. Find the value of A4. e. Let Aŋ be the amount of GLP-1 present after 5n minutes. Find a formula for An. f. Does the sequence {An} converge or diverge? If the : sequence converges, find its limit and explain why this value makes sense in the context of this problem. g. Determine the number of minutes it takes until the amount of GLP-1 in plasma is 1 gram.
Sequences have many applications in mathematics and the sciences. In a recent paper³ the authors write The incretin hormone glucagon-like peptide-1 (GLP-1) is capable of amelio- rating glucose-dependent insulin secretion in subjects with diabetes. How- ever, ‚ its very short half-life (1.5-5 min) in plasma represents a major limita- tion for its use in the clinical setting. The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium. For this exercise, assume the half-life of GLP-1 is 5 minutes. So if A is the amount of GLP-1 in plasma at some time t, then only 4 of the hormone will be present after t + 5 minutes. Suppose A₁ = 100 grams of the hormone are initially present in plasma. a. Let A₁ be the amount of GLP-1 present after 5 minutes. Find the value of A₁. b. Let A₂ be the amount of GLP-1 present after 10 minutes. Find the value of A₂. c. Let A3 be the amount of GLP-1 present after 15 minutes. Find the value of A3. d. Let A be the amount of GLP-1 present after 20 minutes. Find the value of A4. e. Let Aŋ be the amount of GLP-1 present after 5n minutes. Find a formula for An. f. Does the sequence {An} converge or diverge? If the : sequence converges, find its limit and explain why this value makes sense in the context of this problem. g. Determine the number of minutes it takes until the amount of GLP-1 in plasma is 1 gram.
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:7.
Sequences have many applications in mathematics and the sciences. In a recent paper³
the authors write
The incretin hormone glucagon-like peptide-1 (GLP-1) is capable of amelio-
rating glucose-dependent insulin secretion in subjects with diabetes. How-
ever, its very short half-life (1.5-5 min) in plasma represents a major limita-
tion for its use in the clinical setting.
The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium.
For this exercise, assume the half-life of GLP-1 is 5 minutes. So if A is the amount of
GLP-1 in plasma at some time t, then only of the hormone will be present after t + 5
minutes. Suppose A₁ = 100 grams of the hormone are initially present in plasma.
a. Let A₁ be the amount of GLP-1 present after 5 minutes. Find the value of A₁.
b. Let A₂ be the amount of GLP-1 present after 10 minutes. Find the value of A2.
c. Let A3 be the amount of GLP-1 present after 15 minutes. Find the value of A3.
d. Let A4 be the amount of GLP-1 present after 20 minutes. Find the value of A4.
e. Let Aŋ be the amount of GLP-1 present after 5n minutes. Find a formula for An.
f. Does the sequence {An} converge or diverge? If the sequence converges, find its
limit and explain why this value makes sense in the context of this problem.
g. Determine the number of minutes it takes until the amount of GLP-1 in plasma
is 1 gram.
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