MATH*2080W15 - Assignment #2 Initials: Surname: (PRINT) INSTRUCTIONS 1. Print this blank assignment right now. You'll be completing the following 4 pages, writing a summary of your solutions here, and submitting this completed assignment. No extra paper please - I won't accept it. 2. Note that for some questions, there is space provided for summary work. PLEASE DO NOT cram every step into each answer space - but instead - transfer key steps from your rough work. For example, if you prepare a set of equations for solution using Partial Fractions, then show me how you get the equations. Then simply state that after solution the answers are etc. Or, when you perform integration, show me the start, the substitution, the completed set-up, and then say something like, after simplication. Then show me the nal step just before you integrate. 3. IF NOT ALREADY THERE, PLEASE MAKE A BOX AROUND YOUR FINAL ANSWER TO MAKE IT EASIER TO LOCATE AND GRADE (thank you) 4. You are encouraged to work together. PLEASE - PLEASE - write-up your own solutions and submit your own individual write-up. 5. This assignment is worth 13% of your nal grade, graded in 1/2 % units. NOTE: For this assignment, 1) recall 'the drug problem' from MATH1080 (posted on our course link), n and 2) don't forget the closed form of the geometric series 1 + r + r2 + ... + rn = 1r . 1r c S.J. Gismondi (Instructor), 2015. All rights reserved. Assignment #2 MATH*2080W15 Surname: Initials: (PRINT) Student #: Grade: 1. (3 marks) Recall the drug problem from MATH1080 where A(t) = A0 ekt . A drug is administered every 4 hours in doses of 3 mg. The drug is neutralized at an exponential rate with a decay rate constant of k = 0.1. Not all the drug is neutralized before the next dose is administered. Denote by A(t) the amount of drug in the patient's system at time t hours. Compute the sum of the area under each exponential curve for the rst 100 doses (400 hours of treatment), a nite kind of measure of bioavailability. That is, nd an expression for A(t) on each of the rst 100 intervals, integrate each expression over each interval i.e. evaluate each denite integral. Do this say for the rst 3 or 4 expressions and then nd a pattern. Each term is the sum of a series, so make each term into a closed form. Then the sum of these 100 closed forms is also a series that can be made into a closed form. It's a lot of work. But when you are done, you'll have a measure as to how much drug the patient is exposed to, accounting for neutralization over these 100 doses. This is useful when comparing treatments i.e. just divide by 400 hours to get an average amount of drug in the patient's system for comparison purposes. The sum of the area under each exponential curve for the rst 100 doses (in mg*hours) is: c S.J. Gismondi (Instructor), 2015. All rights reserved. 2. (6 marks) A muscle of the Stephantium Gismondious Giganticus hominoid species from Mars is injected with 10mg of cortisone on Monday, Wednesday and Friday at noon on each day, but only for three days. The two compartment model is given by D(t) = D(0)( k2k2 1 )(ek1 t ek2 t ). D(t) is the total amount of k drug in blood serum as a function of t. D(0) is the amount of drug in blood serum immediately after dosing, composed of a 10mg dose and whatever residual may exist. The elimination rate of cortisone from Compartments I and II is k1 = 0.1 and k2 = 0.2 mg per hour respectively.1 (a) State the formula (simplied) for the function D(t) over the indicated intervals (in hours) and compute the residual amount of drug left in the patient's system at the end of each dosing period, just before the next dose is administered. over [0, 48) D(t) = R1 = lim D(t) t48 over [48, 96) D(t) = R2 = lim D(t) t96 over [96, ) D(t) = Compute D(144) Compute lim D(t) t 1 NOTE: The model above DOES NOT model repeated dosing. BUT I still want you to repeat the ideas of the 'drug problem' from MATH1080 (add blood serum levels to existing Compartment I (muscle) drug i.e. residual added to Compartment I) - like we did in class. The model that we should actually use is: D(t) = (D(0) + RI )( k2k2 1 )(ek1 t ek2 t ) + RB ek2 t , where RI is k 'Residual Compartment I' and RB is 'Residual Blood Serum' - too tough for us right now. c S.J. Gismondi (Instructor), 2015. All rights reserved. (b) (i) Suppose that the drug dosage scheme is continued forever i.e. every 48 hours. Write the series that computes the residual amount of drug left in the patient's blood serum at the end of the n'th dosage period, Rn . Use R1 , R2 and D(144) to help you nd the pattern for the series. Next, use your geometric series knowledge from MATH1080 to work out its closed form. Series form of Rn = Closed form of Rn = (c) Finally we're here, the conclusion of the matter. Determine the Eventual Maximum Residual Drug Level (simplied please) i.e. compute the limit of the closed form of Rn as n tends to innity and also the Eventual Maximum Drug Level (also simplied please). Expression for Eventual Maximum Residual Drug Level = Expression for Eventual Maximum Drug Level = 3. (1 mark) Evaluate the improper integral 0 1 4x2 +9 dx. (Perfect please - only one mark e.g. two mistakes will get you 0.) c S.J. Gismondi (Instructor), 2015. All rights reserved. 4. (2 marks) Toxin from a spider bite is neutralized (in an infected body) as a function of time, and the 2 amount of toxin at time t is given by t5 et . Compute the bioavailability of the toxin given by the improper integral 2 t5 et dt i.e. toxin enters the body at time t = 0. 0 1 5. (1 mark) Evaluate the limit of x x as x tends to innity. c S.J. Gismondi (Instructor), 2015. All rights reserved. Solution 1: Since the initial dose is A0, the drug concentration at any time t o is found by the equation A=A0e-kt At t=T the second dose of A0 is taken, which increases the drug level to A(T)=A0+A0 e-kT = A0(1+e-kT) The drug level immediately begins to decay. To find its mathematical expression we solve the initial-value problem: dA kA dt A(T)=A0(1+e-kT) Solving this initial value problem we get A=A0(1+e-kT)e-k(t-T) This equation gives the drug level for t>T. The third dose of A0 is to be taken at t=2T and the drug just before this dose is taken is given by A Ao 1 e kT e k ( 2T T ) Ao (1 e kT )e kT The dosage y0 taken at t=2T raises the drug level to A(2T) = A0 + A0(1+e-kT)e-kT = A0(1+e-kT+e-2kt) Continuing in this way, we find after (n+1)th dose is taken that the drug level is A(nT)=A0(1+e-kT+e-2kT+.....+e-nkT) We notice that the drug level after (n+1)th dose is the sum of the first n terms of a geometric series, with first term as yo and the common ratio e-kT. This sum can be written as Ao (1 e ( n 1) kT ) A(nT ) 1 e kT Now putting the values, n =99, Ao = 3mg , k =0.1 , T =4 hours After computing, A(99T) = 9.099 = 9.10 mg \f\f\f