Assign.#1Sol.

# Assign.#1Sol. - MAT 1330: Calculus for the Life Sciences I...

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MAT 1330: Calculus for the Life Sciences I 01.10.2008 Assignment 1, SOLUTIONS Pawel Lorek University of Ottawa Problem 1: [4 points] Suppose that every morning a patient receives the same dose of drug. From the dose, the drug concentration in his blood increases by 2. Over the course of 24 hours between doses, 75% of the drug in the blood is removed. (a) Write the linear DTDS for the drug concentration, x t +1 = f ( x t ) , and ±nd x 4 when x 0 = 88 . (b) Draw the updating function and start the cobwebbing process at x = 0 . 2 . (c) Find the equilibrium explicitly. (d) Is the equilibrium stable? Use the stability criterion from class and compare with your cobwebbing. Solution Because formulation of the Problem was ambiguous two solutions are accepted: with DTDS x t +1 = 0 . 25 x t + 0 . 5 and with DTDS x t +1 = 0 . 25 x t + 2 Solution with x t +1 = 0 . 25 x t + 0 . 5 (a) Just after receiving the medication the concentration is x n + 2, but after 24 hours, before next dose is applied, 75% of drug is removed, i.e. we have x n +1 = ( x n + 2) 0 . 75( x n + 2) = 0 . 25 · x n + 0 . 5 We have x 0 = 88 x 1 = 0 . 25 · x 0 + 0 . 5 = 0 . 25 · 88 + 0 . 5 = 22 . 5 x 2 = 0 . 25 · x 1 + 0 . 5 = 0 . 25 · 22 . 5 + 0 . 5 = 6 . 125 x 3 = 0 . 25 · x 2 + 0 . 5 = 0 . 25 · 6 . 125 + 0 . 5 = 2 . 03125 x 4 = 0 . 25 · x 3 + 0 . 5 = 0 . 25 · 2 . 03125 + 0 . 5 = 1 . 0078125 (b) See Figure 1. (c) Equilibrium x * must ful±lls x * = f ( x * ) i.e. x * = 0 . 25 · x * + 0 . 5 substract 0 . 25 · x * from both sides 0 . 75 · x * = 0 . 5 divide both sides by 0.75 x * = 0 . 5 0 . 75 = 1 2 · 4 3 = 2 3 (d) The slope of updating function is equal to 1 4 . And | 1 4 | < 1, so the Slope Criterion implies that x * = 2 3 is a stable equilibrium. Cobwebbing, we should obtain the same equilibrium starting at x 0 n = 0 . 2. 1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x n x n+1 x n+1 = 0.25 x n + 0.5 y = x y=f(x) = 0.25 x+0.5 Figure 1: Problem 1: Cobwebbing with x 0 = 0 . 2 Solution with x t +1 = 0 . 25 x t + 2 The analysis is similar, here are just results: (a) x 0 = 88 , x 1 = 24 , x 2 = 8 , x 3 = 4 , x 4 = 3, (b) similar, (c) x * = 8 3 , (d) the same. 2
Problem 2: [4 points] A group of patients is given a certain dose of a drug once. Two measurements of concentration of the drug in the blood are taken 24 hours apart to de- termine the rate at which the drug is removed from the blood stream. The measurements are given below. patient initial measurement Fnal measurement 1 3 1 2 4.5 1.5 3 0.6 0.2 4 1.8 0.6 (a) Write a DTDS of the form x t +1 = ax t for drug removal and Fnd the value of a. (b) ±or patient 1, how long will it take until the drug concentration is below 0.1?

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## This note was uploaded on 01/25/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Spring '08 term at University of Ottawa.

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Assign.#1Sol. - MAT 1330: Calculus for the Life Sciences I...

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