MATA 2634
Department of Mathematics and Applied Mathematics
Departement van Wiskunde en Toegepaste Wiskunde
Semester Test 1 / Semestertoets 1
2016
Answer all questions and show all your calculations. /
Beantwoord alle vrae en toon alle berekeninge
Punte/Marks 45
Question 1/ Vraag 1
(3)
1. State whether the following is true or false and give a reason.
1. Dui aan of die volgende waar of vals is en gee ’n rede.
(a) The dynamical system Sn+1 = 34 Sn for all n ≥ 0 is equivalent to the dynamical system Tm−1 = 43 Tm for all m ≥ 1.
(a) Die dinamiese stelsel Sn+1 = 43 Sn vir alle n ≥ 0 is ekwivalent aan die dinamiese stelsel Tm−1 = 43 Tm vir alle m ≥ 1.
Question 2/ Vraag 2
(12)
Consider the following sequences:
Beskou die volgende rye:
(an ) = (1, 4, 10, 19, 31, . . . ) and/en (bn ) = (3, 5, 11, 29, . . . ) ; n ≥ 0
(a) Determine the next three terms of (an ) and (bn ).
(a) Bepaal die volgende drie terme van (an ) en (bn ).
1−3n+1
1−3 .
n+1
n
i
Σi=0 3 = 1−3
1−3 .
(b) Find the nth term of (bn ). Use that Σni=0 3i =
(b) Vind die n
de
term van (bn ). Gebruik dat
Question 3/ Vraag 3
(6)
(6)
(13)
Suppose that a contaminated body of water is being cleaned by a filtering process. Each week this process is capable of filtering out a certain fraction a
(where 0 < a < 1)) of the concentration of the pollutants. Assume that another
amount b of pollutants seep in. Let pn denote the mass(in tons) at the end of
week n.
Veronderstel dat gekontamineerde water deur ’n filteringsproses gesuiwer word.
Die proses verwyder elke week ’n fraksie a (waar 0 < a < 1) van die besmette
stof. Neem ook aan dat daar elke week b van die besmette stof bygevoeg word.
Laat pn die massa(in ton) van die besmette stowwe aan die einde van week n
wees.
1
(a) Formulate a recurrence relation to describe the mass of the pollutants after
week n + 1 in terms of the mass of the pollutants after n weeks for all n ≥ 0.
(a) Formuleer ’n rekursie relasie om die massa van die besmette stof in die water
na n + 1 weke te beskryf in terme van die massa van die besmette stowwe na n
weke vir alle n ≥ 0.
(3)
(b) Assume that 5% can be removed each week. Find the value of a.
(2)
(b) Neem aan dat 5% elke week verwyder kan word. Vind die waarde van a.
(c) If 2 tons of the pollutants seep in every week and p0 = 8 tons, find a formula
for pn .
(c) Indien 2 ton van die besmette stof elke week bygevoeg word en p0 = 8 ton,
vind ’n formule vir pn .
(5)
(d) Find the equilibrium point(s) if they exist.
(d) Vind die ekwilibriumpun(e) indien dit bestaan.
(3)
(5)
Question 4/ Vraag 4
Consider the following table:
Beskou die tabel:
y 3.5 5 6 7
8
x
3
6 9 12 15
1
Using the fact that y ∝ x 2 is a good assumption, find a value for k such that
1
y ≈ kx 2 .
1
1
Neem aan dat y ∝ x 2 ’n goeie aanname is en vind ’n waarde vir k sodat y ≈ kx 2 .
Question 5/ Vraag 5
(15)
Suppose that a certain chronic illness has three stages of severity, stages I, II
and III. If someone is in stage I, there is a 10% chance to progress to stage
II and no chance to progress to stage III. If someone is in stage II, there is a
10% chance to progress to stage III and a 10% chance to return to stage I. If
someone is in stage III, there is a 20% chance to return to stage II. Otherwise
a person remains in their current stage.
Veronderstel dat ’n kroniese siekte oor drie vlakke van erns onderskei, vlakke
I, II en III. Indien ’n persoon in vlak I is, is die kans om na vlak II toe te
vorder 10%. Indien ’n persoon in vlak II is, is die kans om na vlak III te vorder
10% en ook om na vlak I terug te keer 10%. Indien ’n persoon in vlak III is,
is die kans om terug te keer na vlak II, 20%. Andersins bly ’n persoon in die
betrokke vlak.
(a) Draw a diagram to indicate the dynamics of the stages of the chronic illness,
letting stage I, II and III be denoted by F, S and T respectively.
(a) Teken ’n diagram om die dinamika van die vlakke van die kroniese siekte,
deur vlak I, II en III deur F, S en T onderskeidelik aan te dui, te beskryf.
2
(6)
Fn+1 = 0.9Fn + 0.1Sn + 0.0Tn
Sn+1 = 0.1Fn + 0.8Sn + 0.2Tn
Tn+1 = 0.0Fn + 0.1Sn + 0.8Tn
If 5000 patients are treated at a facility, show that this reduces to:
Indien 5000 pasiënte by ’n fasiliteit behandel word, wys dat dit reduseer na:
Fn+1 = 0.9Fn + 0.1Sn
(4)
Sn+1 = −0.1Fn + 0.6Sn + 1000
(b) Assume that:
(b)Neem aan dat
(c) Find the equilibrium point. Give the answer as a 3-tuple (F, S, T )
(c) Find the equilibrium point. Skryf jou antwoord as ’n 3-tal (F, S, T )
(5)
Total Marks/Totale Punte: 48
3