Department of Mathematics and Applied Mathematics
MAM115
Final Assignment
Due date: 30 June 2021
Full marks: 100
This is the final assignment for MAM115. To be able to answer the questions you will need to
know work from the whole semester. The questions also require you to think critically about
numbers and functions and show that you understand the calculations that you make.
• It is not necessary to type the assignment. You may submit a scan or photo of neatly
handwritten work.
• Show all your calculations and workings in each question. You will need to use
a calculator at times. When your answer is not a whole number, or a quotient of whole
numbers, round off to two decimal places.
• You will be given marks for your solutions as well as how well you set out your work and
the clarity and originality in your answers.
• You may consult other sources for assistance. If you do so, please reference these clearly.
The university policies for copying and plagiarism will apply.
• The deadline is 11:55pm (5 minutes before midnight) on 30 June 2021 but do not start
the assignment on the last day – it will take some time to think about the questions and
write your answers clearly.
1. In this question we have three functions:
• f (t) is the number of UWC students who register for MAM115 in year t.
• g(t) is the number of UWC Pharmacy students who register for MAM115 in year t.
• h(x) is the number of MAM115 students who pass MAM115 if there are x students
in the class.
(a) Consider the following 10 combinations of these functions and identify the combination that best measures the quantity given in each of (i) to (v) below.
g(t)(h ◦ f )(t)
h(x)
(f ◦ h)(t)
(f ◦ h)(t);
(h ◦ f )(t);
;
;
1−
;
f (t)
f (t)
f (t)
f (t) − g(t)
f (t) − (h ◦ f )(t)
;
; 1 − h(f (t) − 1); f (t − 1) − (h(f (t − 1)); h(x) − g(t).
f (t)
f (t)
i. The number of students who pass MAM115 in year t.
ii. The failure rate for MAM115 in year t.
iii. The number of students who failed MAM115 one year ago.
iv. The proportion of students in the MAM115 class who are not doing Pharmacy.
v. The number of Pharmacy students who pass MAM115 in year t assuming that
they have the same pass rate as the rest of the class.
[8]
(b) If you are told that h0 (x) < 0, would you recommend that more or fewer students
should register for MAM115? Explain.
[3]
(c) If h0 (f (t))f 0 (t) > 0, what does that say about MAM115 over time? Explain.
[4]
(15 marks)
2. Unfortunately many people have died in South Africa due to the COVID-19 pandemic.
The black graph below shows the number of people over age 1 year who have died per
week (from all possible causes, not only COVID-19) in South Africa between 29 December
2019 and 12 June 2021. We will call this black graph D(t). The solid orange graph shows
how many people are normally expected to die per week, carrying on to the end of 2021.
We will call this graph E(t). (The dotted lines are just to show that the actual values may
be a little different to E(t).)
[Downloaded from https://www.samrc.ac.za on 16.06.2021.]
The functions f (t) and g(t) have similar shapes to D(t) and E(t) respectively. They are
sketched below on the interval [0, 48], measuring t in weeks and taking t = 0 on 1 June
2020.
f (t) =
a 5 23 4 3945 3 8695 2
t − t +
t −
t + 7176t + b
256
16
64
8
g(t) = 2000 cos
πt
30
+ 8500
(a) Are D(t) and E(t) functions or not? Explain your answer.
[2]
(b) Solve for a and b if the maximum value of f (t) on the interval [0, 48] is at the point
(32; 24900).
[7]
(c) At what week in the interval is g(t) at its minimum?
[6]
(d) At what week in the interval is g(t) increasing the fastest?
Z 48
(e) Calculate the integral
(f (t) − g(t))dt.
[6]
[9]
0
(f) Explain clearly what the units are for your answer in (e) and what it represents. [4]
(g) In a short paragraph or two explain why the integral in (e) may be the best way to
calculate the impact of the COVID-19 pandemic on people’s lives.
[6]
(40 marks)
3. One of the ways society is trying to bring the pandemic under control is by vaccinating
people against the virus. The graph below shows the number of people vaccinated in South
Africa per day from the end of April to early in June. It is clear that the number of people
being vaccinated per day has increased significantly over the past two months.
[Downloaded from https://www.statista.com/statistics/1231031/daily-number-of-covid-19vaccination-doses-in-south-africa/ on 16.06.2021.]
On 1 May 2021, 3,000 people were vaccinated and on 31 May 2021, 45,000 people were
vaccinated.
(a) Setting t = 0 on 1 May and t = 30 on 31 May, you want to find a function whose
graph looks similar to the shape above and has the same values of 3,000 at t = 0 and
45,000 at t = 30. In each of (i) to (iv) below solve for the constants a and b so that
the given function goes through the above points at t = 0 and t = 30.
i. f (t) = at2 + b
ii. g(t) = aebt
π iii. h(t) = a + b cos
t
40
a
iv. j(t) =
[18]
1 + be−t/10
(b) For each of the four functions from (i) to (iv) in (a) calculate the value of the derivative
when t = 30. (Use the values for a and b that you calculated in (a).)
[10]
(c) Based on your answers in (b), if you were the new Minister of Health, which function
would you like to see as the function describing the number of people vaccinated per
day? Explain.
[4]
(d) Using the graph given above, draw a rough sketch showing the total number of people
vaccinated since 1 May 2021 against time. Draw it on the same domain as the graph
above and explain clearly the reasons for the slope and concavity of your graph. [4]
(e) Your friend says that she expects that if you carry on beyond June 2021, the graph
of the total number of people vaccinated since 1 May 2021 will change concavity at
some point in the future. Do you agree with her or not? Explain.
[4]
(f) Only one of the functions (i) to (iv) above will give a graph of the total number of
people vaccinated from 1 May 2021 to 1 May 2022 that changes concavity. Which
function is that? Explain.
[5]
(45 marks)