STAT3110 STAT6110 Statistical Inference/COMP4500 COMP7500 Advanced Algorithms and Data Structures/ER Modelling/ENM1600 Engineering Mathematics

STAT3110 STAT6110 Statistical Inference

Due at 11:50pm on Sunday 21st August, 2022.

1. [18 marks] A random variable X has probability density function f(x) = cx for
0 < x <
1
2
and f(x) = 0 otherwise, where c is a constant.
(a) Show that constant c = 8 defines pdf properly.
(b) Show that µ = E(X) =
1
3
and σ 2 = Var(X) =
1
72 .
(c) If X 1 , X 2 , ... are independent and identically distributed with the same distribu-
tion as X, then use the Central Limit Theorem to find an approximation to
Pr
? 288
X
i=1
X i ≥ 40
?
.
You may express your answer in terms of the standard Normal cumulative distri-
bution function Φ.
(d) Find Var(X 2 ).
(e) Show that
1
n
n
X
i=1
X 2
i
P
−→
1
8
as n −→ ∞

2. [16 marks] Suppose that two independent random samples of size n 1 and n 2 observa-
tions are obtained. Let X 1 ,X 2 ,...,X n 1 and Y 1 ,Y 2 ,...,Y n 2 be the two random samples
and suppose that E(X i ) = E(Y i ) = λ and Var(X i ) = Var(Y i ) = σ 2 < ∞. The sample
means from each sample are as follows,
X =
P n 1
i=1 X i
n 1
, Y =
P n 2
i=1 Y i
n 2
.
Also defined two pooled estimators,
L 1 =
n 1 X + n 2 Y
n 1 + n 2
and L 2 =
1
2
? X + Y ?
.
(a) Show that both of the pooled estimators E(L 1 ) = E(L 2 ) = λ.
(b) Find Var(L 1 ) and Var(L 2 ).
(c) Show that both estimators are consistent estimators of λ.
(d) Derive the efficiency of the estimator L 1 relative to L 2 .
(e) State which estimator is preferable and provide a reason.

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COMP4500 COMP7500 Advanced Algorithms and Data Structures

3:00pm, Friday the 16th September 2022

Part A (25 marks total)
Question 1: Constructing SNI and directed graph [5 marks total]
(a) (1 mark) Creating your SNI. In this assignment you are required to use your student number to
generate input.
Take your student number and prefix it by “98” and postfix it by “52”. This will give you a twelve
digit initial input number. Call the digits of that number d[1],d[2],...,d[12] (so that d[1] = 9,d[2] =
8,...,d[12] = 2).
Apply the following algorithm to these twelve digits:
1 for i = 2 to 12
2 if d[i] == d[i − 1]
3 d[i] = (d[i] + 3) mod 10
After applying this algorithm, the resulting value of d forms your 12-digit SNI. Write down your initial
number and your resulting SNI.
(b) (4 marks) Construct a graph S with nodes all the digits 0,1,...,9. If 2 digits are adjacent in your
SNI then connect the left digit to the right digit by a directed edge. For example, if “15” appears in
your SNI, then draw a directed edge from 1 to 5. Ignore any duplicate edges. Draw a diagram of the
resulting graph. (You may wish to place the nodes so that the diagram is nice, e.g., no or few crossing
edges.)
Question 2: Strongly connected components [20 marks total]
Given a directed graph G = (V,E), a subset of vertices U (i.e., U ⊆ V ) is a strongly connected component
of G if, for all u,v ∈ U such that v 6= u,
a) u and v are mutually reachable, and
b) there does not exist a set W ⊆ V such that U ⊂ W and all distinct elements of W are mutually
reachable.
For any vertices v,u ∈ V , v and u are mutually reachable if there is both a path from u to v in G and a
path from v to u in G.
The problem of finding the strongly connected components of a directed graph can be solved by utilising
the depth-first-search algorithm. The following algorithm SCC(G) makes use of the basic depth-first-search
algorithm given in lecture notes and the textbook, and the transpose of a graph; recall that the transpose of
a graph G = (V,E) is the graph G T = (V,E T ), where E T = {(u,v) | (v,u) ∈ E} (see Revision Exercises 3:
Question 6). (For those who are interested, the text provides a rigorous explanation of why this algorithm
works.)
SCC(G)
1 call DFS(G) to compute finishing times u.f for each vertex u
2 compute G T , the transpose of G
3 call DFS(G T ), but in the main loop of DFS, consider the vertices in order of decreasing u.f
4 output the vertices of each tree in the depth-first forest of step 3 as a separate
strongly connected component

(a) (10 marks) Perform step 1 of the SCC algorithm using S as input. Do a depth first search of S (from
Question 1b), showing colour and immediate parent of each node at each stage of the search as in
Fig. 20.4 of the textbook [4th edition]. That means that you should draw the annotated graph for
each stage of the search. (We want to make sure that you understand how the algorithm works.) Also
show the start and finish times for each vertex.
For this question you should visit vertices in numerical order in all relevant loops:
for each vertex u ∈ G.V and
for each vertex v ∈ G.Adj[u].
(b) (2 marks) Perform step 2 of the SCC algorithm and draw S T .
(c) (8 marks) Perform steps 3, 4 of the SCC algorithm. In your solution you must list (and draw) the
trees in the depth-first forest in the order in which they were constructed. (You do not need to show
working.)
Part B (75 marks total): Meeting preparation
[Be sure to read through to the end before starting.]
An important aspect of the job of prime minister is meeting with people.
There are a set of people, P, that the prime minister is able to meet with. The prime minister can only
meet with one person at a time, and each meeting takes exactly one unit of time. Each meeting is therefore
uniquely identified by a person p ∈ P and a time t ∈ Z ≥0 (where Z ≥0 is the set of integers that are greater
than or equal to zero).
Each person p ∈ P has a set of times that they are available to meet with the prime minister. Given the
availability of the people, the set of all meetings that are possible is described by the set availableMeetings,
where a meeting (p,t) is in availableMeetings if and only if person p is available to meet with the prime
minister at time t.
For both spatial and political reasons, the order of meetings is constrained in the following way. After
meeting with a person p ∈ P, the prime minister’s next meeting, if any, must be with a person in the set
next(p), where next ∈ P → PP is a total function from P to the power set of P. If, for example, next(p) is
the empty set, then following a meeting with person p, no further meetings are possible.
A meeting schedule is a finite list of one or more meetings,
h(p 0 , t 0 ), (p 1 , t 1 ), ... , (t n , t n )i
such that:
• All of the meetings in the schedule are possible given the availability of the people. That is, for all
i ∈ {0,1,...,n} we have that (p i ,t i ) ∈ availableMeetings;
• The meetings in the schedule are ordered by time and do not overlap. That is, for all i ∈ {0,1,...,n−1}
we have that t i < t i+1 ;
• The order of meetings respects the ordering described by the function next. That is, for all i ∈
{0,1,...,n−1} we have that p i+1 ∈ next(p i ).

The duration of a schedule is defined to be the meeting time of the last meeting in the schedule minus the
meeting time of the first meeting in the schedule plus one, i.e. (t n −t 0 )+1. For example, the duration of a
schedule with only one meeting is 1.
The prime minister has told a staffer to prepare a schedule for him for the next day in which the first meeting
is with person first ∈ P, the last meeting in the schedule is with person last ∈ P, and the schedule has the
smallest duration of any schedule that satisfies that requirement. For brevity, we say that any schedule in
which the first meeting is with person first, the last meeting is with person last, and the duration of the
schedule is less than or equal to the duration of any schedule that satisfies that requirement, is optimal with
respect to first and last. (Note that there may be zero or more optimal schedules w.r.t. first and last.)
The prime minister would like to prepare for his schedule by briefing himself on all of the people that he will
meet the next day. Unfortunately, the staffer has gone home early without providing him with the schedule,
and so the prime minister needs to work out for himself the set of all people that he might be meeting with
the next day.
That is, the prime minister would like to find the set mightMeet of people such that p ∈ mightMeet if and
only if there exists one or more optimal schedules (w.r.t. first and last) that contain one or more meetings
with person p. Note that this means that if there are no optimal schedules (w.r.t. first and last) then the
set mightMeet will be empty (i.e. the staffer may not have been able to prepare a schedule that satisfies
the prime minister’s requirements).
Example 1 For example consider a situation in which P = {p0,p1,p2,p3,p4,p5,p6,p7,p8}, first = p3,
last = p5 and
next = {p0 7→ {p1,p3},
p1 7→ {p2},
p2 7→ {p5},
p3 7→ {p1,p4,p6},
p4 7→ {p5,p8},
p5 7→ {},
p6 7→ {p7},
p7 7→ {p4}
p8 7→ {p7}} ,
and
availableMeetings = {(p0, 3), (p0, 4),
(p1, 1),
(p2, 5), (p2, 10),
(p3, 0), (p3, 2), (p3, 7), (p3, 10),
(p4, 7), (p4, 13), (p4, 16),
(p5, 5), (p5, 7), (p5, 12), (p5, 17),
(p6, 3), (p6, 9),
(p7, 4), (p7, 15),
(p8, 14)} .
For this example there are four optimal schedules with respect to first and last:
h(p3, 0), (p1, 1), (p2, 5), (p5, 7)i
h(p3, 10), (p4, 13), (p5, 17)i
h(p3, 10), (p4, 16), (p5, 17)i
h(p3, 10), (p4, 13), (p8, 14), (p7, 15), (p4, 16), (p5, 17)i
This means that the prime minister needs to prepare for meetings with all of the people in the set
mightMeet = {p1,p2,p3,p4,p5,p7,p8}. The optimal schedules each have a duration of 8. There are other

schedules that start with a meeting with p3 and end with a meeting with p5, but they have longer durations
(and are therefore not optimal). For example, a schedule with a longer duration of (12 − 2) + 1 = 11 is:
h(p3, 2), (p6, 3), (p7, 4), (p4, 7), (p5, 12)i .
Your task is to design, implement and analyze an algorithm that takes as input:
• a mapping next from each of the people p ∈ P to the set of people who can be met with directly after
person p,
• a list availableMeetings of the meetings that are possible given the availability of the people p ∈ P,
• a person first ∈ P,
• a person last ∈ P
and returns the set mightMeet of people such that p ∈ mightMeet if and only if there exists one or more
optimal schedules (w.r.t. first and last) that contain one or more meetings with person p. For simplicity
you may assume that the list of available meetings availableMeetings is sorted in ascending order of meeting
time. (Note that in the example listed above, availableMeetings is not presented in sorted ascending order
of meeting time for readability reasons – instead the presentation is grouped by person. In the input to your
algorithm (see test handoutTest1 from FinderTest.java for an example) the list availableMeetings will
be sorted in ascending order of meeting time.)
You algorithm must be designed and implemented as efficiently as possible.
Question 3: Design and implement an efficient solution (50 marks)
Design and implement an algorithm that answers the question above. Your algorithm should be as efficient
as possible. Marks will be deducted for inefficient algorithms (e.g. a brute-force approach is not appropriate).
Clearly structure and comment your code. Use meaningful variable names.
• Your algorithm should be implemented in the static method Finder.findPeople from the Finder
class in the assignment1 package that is available in the zip file that accompanies this handout.
The zip file for the assignment also includes some other code that you will need to compile the class
Finder as well as some junit test classes to help you get started with testing your code.
• Do not modify any of the files in package assignment1 other than Finder, since we will test your code
using our original versions of these other files.
• You may not change the class name of the Finder class or the package to which it belongs. You
may not change the signature of the Finder.findPeople method in any way or alter its specification.
(That means that you cannot change the method name, parameter types, return types or exceptions
thrown by the method.)
• Your implementation should be in Java 1.8. You are encouraged to use Java 8 SE API classes, but no
third party libraries should be used. (It is not necessary, and makes marking hard.)
• Don’t write any code that is operating-system specific (e.g. by hard-coding in newline characters etc.),
since we will batch test your code on a Unix machine. Your source file should be written using ASCII
characters only.
• You may write additional classes, but these must belong to the package assignment1 and you must
submit them as part of your solution – see the submission instructions for details.

• The junit test classes as provided in the package assignment1.test are not intended to be an exhaus-
tive test for your code. Part of your task will be to expand on these tests to ensure that your code
behaves as required.
Your implementation will be evaluated for correctness and efficiency by executing our own set of junit
test cases. Code that is submitted with compilation errors, or is not compatible with the supplied testing
framework will receive 0 marks. A Java 8 compiler will be used to compile and test the code.
Hint: It may be very inefficient to solve the problem by calculating all of the possible schedules
that start with a meeting with first and end with a meeting with last. The problem can be
solved in polynomial time! Algorithms similar to the ones you are already familiar with from
class could be adapted to solve the problem, or inspire a solution.
Question 4: Worst-case analysis (25 marks)
This question involves performing an analysis of the worst-case time complexity and worst-case space com-
plexity of your algorithm from Question 3.
(a) (5 marks) For the purpose of the worst-case time complexity analysis in Q4(b) and the worst-case space
complexity analysis in Q4(c), provide clear and concise pseudocode that summarizes the algorithm
you used in your implementation from Question 3. You may use the programming constructs used in
Revision solutions, and assume the existence of common abstract data types like sets, maps, queues,
lists, graphs, as well as basic routines like sorting etc.
Clearly structure and comment your pseudocode. Use meaningful variable names.
[It should be no more than three pages using minimum 11pt font. Longer descriptions will not be
marked.]
(b) (12 marks) Let m be the number of meetings in availableMeetings and let n be the total number of
people (i.e. the number of people in the domain of mapping next).
Provide an asymptotic upper bound on the worst case time complexity of your algorithm in terms
of parameters m and n. Make your bound as tight as possible and justify your solution using your
pseudocode from Q4(a).
You must clearly state any assumptions that you make (e.g. on the choice of implementations of any
data structures that you use, and their running time etc.).
To simplify your analysis, you must make the (incorrect but simplifying) assumption that HashSet
and HashMap operations that have expected-case time complexity O(1) actually have worst-case time
complexity O(1). E.g. checking for set-membership in a HashSet has expected-case time complexity
that is O(1).
[Make your analysis as clear and concise as possible – it should be no more than a page using minimum
11pt font. Longer descriptions will not be marked. Also note that to receive any marks for this question,
you must justify your solution – it is not enough to only give an asymptotic upper bound without
explanation.]
(c) (8 marks) As for Q4(b), let m be the number of meetings in availableMeetings and let n be the total
number of people (i.e. the number of people in the domain of mapping next).
Provide an asymptotic upper bound on the worst case space complexity of your algorithm in terms
of parameters m and n. Make your bound as tight as possible and justify your solution using your
pseudocode from Q4(a).
You must clearly state any assumptions that you make (e.g. on the choice of implementations of any
data structures that you use, and their space usage etc.).
[Make your analysis as clear and concise as possible – it should be no more than a page using minimum
11pt font. Longer descriptions will not be marked. Also note that to receive any marks for this question,
you must justify your solution – it is not enough to only give an asymptotic upper bound without
explanation.]

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ER Modelling

Commonwealth Games

The Commonwealth Games are a series of sporting events held every 4 years, which host competitors from a body of 72 member nations and territories, including Australia. The 2022 Commonwealth games have just ended, and the organisers are looking ahead towards the 2026 Games, which are to be held here in Victoria.

Your team is going to be helping to create the technical infrastructure for recording the athletes, events, and results of the games. As part of this, you’ll be creating a Conceptual ER Model in Chen’s notation to store this information. The following specifications have been provided to you to assist in your design. The team also wants to store data about future games, and possibly previous ones too so that it can all be kept in the one system and accessed conveniently.

Note: Some simplifications have been made compared to the real games, in particular we do not carefully consider round-based sports (e.g. basketball), do not consider qualifier data, etc. If at any point your understanding of The Games and this specification contradict, consider this specification as authoritative.

Regions, countries, and Games

Currently, 72 member nations (countries) compete in The Games. These nations are divided into 6 fixed regions: Africa, Americas, Caribbean, Europe, Asia, and Oceania. For each member nation we should store their name, and a link to an image of their flag.

The Games are held every 4 years, and are located in one of the member nations. Each set of games has a ‘host city’ within that nation (e.g. Melbourne, Australia). Each set of games has its own ‘theme song’ for which we need to store the name (e.g. ‘This is the Moment’), as well as a music video recording (to be played at the event).

Sports, Events, and Competitors

The Games are broken down into various sports (e.g. ‘swimming’, ‘basketball’, ‘shotput’, ‘wheel-chair race’, etc.). Each sport consists of a set of events (e.g. ‘100m men’s breast-stroke’ and ‘50m women’s breast-stroke’ are both events that fall under the sport of ‘swimming’). Each event is for only one sport. New sports and events may be added, or old ones removed, from future games. Events may be either ‘time-based’ or ‘scorebased’. Additionally, all events have a set of rules (which can be 10,000+ chars long) which we should store.

Events are competed in by ‘competitors’. Events can be ‘individual’ or ‘group’ based. In individual events, each ‘competitor’ is a single ‘athlete’ (e.g. ‘100m men’s sprint’). For group events, each ‘competitor’ is actually a group of several athletes that compete together (e.g. a relay race). Every competitor in The Games competes as a representative of one of the member nations.

Note that each country might be represented by several competitors in the same event. This is true for both group and individual events. For example, out of 10 swimmers in the ‘100m men’s breast-stroke’ (which is an individual event), there might be 2x Australian athletes who are competing separately and are thus separate competitors.

For each athlete, we need to record their name (which consists of their first, middle, and last name, and their title), and date of birth. Athletes may compete in many different games over time, and might potentially represent more than one country over time.

As part of each event, we may store up to 1 photo per competitor that was taken during the event. These photos will have been uploaded online, and so we only store the link. For instance, for the ‘100m men’s breaststroke’ (an individual event), we might store a link for up to one photo that was taken during the event of each of the individual swimmers. For a group event like the ‘1000m men’s freestyle relay’, where the ‘competitor’ is in fact 4 athletes, we still only store up to one photo per competitor/group of 4.

Scores and Medals

For each competitor, we need to record their ‘performance’ in the event. Each event can be one of the following types depending on what performance means in that context:

1. Time based events have a final time that the competitor took to perform a task (like swimming, running), as well as a ‘penalty time’ which may be used if the competitor was penalised during the event (i.e. broke an event rule).

2. Score based events have a final score that the competitor earned (like diving, archery). They also have a field for ‘disqualification’ set to ‘true’ if the competitor was disqualified for any reason. Score-based events may also have been scored by a panel of judges (e.g. in diving), in which case we need to record the names of all judges on the panel for that event. The same judges will judge all competitors in the one instance of the event (e.g. the same judges assessed all of the competitors in the ‘2022 men’s springboard diving’)

Note that we record times/scores per ‘competitor’, so for group-based events we’d record only a single ‘time’ or ‘score’ record for the group as a whole.

For all event types, we should also record the date and time that each event occurred, and the number of spectators present. For some events the average and maximum outside temperature recorded during that event must be recorded. Gold, Silver, and Bronze medals are awarded based on competitors’ performances.

Venues

Each event takes place in a venue provided by the host city. For each venue, we need to store the name, latitude + longitude, spectator capacity, and a photo of the venue.

Some venues might be ‘inside’ of other venues. For example, a swimming event might be conducted in the ‘Queen Elizabeth Swimming Pool’, but this pool is a part of a larger venue called the ‘London Aquatics Center’. Note that some other event might just be listed as being held at the ‘London Aquatics Center’, rather than another venue inside of it.

Business Requirements

Your database design needs to be able to meet the business' needs to answer questions such as:

1. Which series of games did competitor ‘Emma McKeon’ participate in?

2. List names of the events that belong to a sport ‘swimming’ but have never been held at The Games.

3. List the names of all events that were held in the 1974 games.

4. Find the youngest Badminton player that has ever represented Bermuda.

5. Which athlete had the lowest time for the 2022 games’ Women’s Wheelchair Race event (and thus won the gold medal)?

6. Which venue has the greatest number of venues inside of it?

7. How many sports have had events in the past, but had no events in the 2022 games?

8. Which country has had the greatest number of competitors across all games?

9. What were the names of the judges that scored the Men’s Diving event at the 2018 games?

10. Which sport’s events had the highest average number of spectators?

11. What is the record keeping time in 100m men’s breast-stoke across all the games recorded?

12. How many competitors were disqualified or received penalty time in the 2018 games?

13. Is there a link to a photo of athlete ‘Kyle Chalmers’ competing in the 2022 games in the event ‘men’s 100m freestyle’?

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ENM1600 Engineering Mathematics

Friday 9 September 2022

QUESTION 1 (55 marks)
The currents I 1 , I 2 , and I 3 , measured in Amperes, in the three branches of a parallel circuit are
determined by the following system of linear equations.
3I 1 − 5I 3 = −16
9I 1 − 7I 2 − I 3 = −48
−6I 1 + 9I 2 = 42
(a) Write the system in matrix form Ax = b; (15 marks)
(b) Solve the system of linear equations using Cramer’s Rule. (40 marks)
Check your answer by substituting into the system of linear equations.
QUESTION 2 (20 marks)
The steady-state concentrations of chemicals in three vats C 1 , C 2 , and C 3 , measured in milligrams
per cubic metre (mg/m 3 ), are determined by the following system of linear equations.
−3C 1 + 2C 2 − 4C 3 = −8
6C 1 + 20C 2 − 13C 3 = 118
−9C 1 + 10C 2 − 17C 3 = −10
(a) Write the system of linear equations in augmented matrix form. (12 marks)
(b) Use Gaussian elimination on the augmented matrix to find the three concentrations, and
write the row operation used next to each new row. (40 marks)
Check your answer by substituting into the system of linear equations.
QUESTION 3 (28 marks)
Let f(x) = 5ln(3x + 6) and g(x) = 1 + 3cos(6x).
(a) Find the composite function f(g(x)) and give its domain (i.e. the values of x for which the
composite function is defined). (14 marks)
(b) Find the composite function g(f(x)) and give its domain (i.e. the values of x for which the
composite function is defined). (14 marks)
QUESTION 4 (25 marks)
If g(x) is any (nonzero) odd function of x, determine whether the following functions,
f(x) =
? 3 + g(x) ? 2 ,
f(x) = cos
? g(x) ? ,
f(x) = e g(x) − e −g(x) ,
are even, odd functions of x, or none of these.

QUESTION 5 (40 marks)
The following rational functions were found in analyzing the motion of a spring-damper system.
Express these rational functions as partial fractions:
(a)
30s + 23
18s 2 + 51s + 35 ;
(20 marks)
(b)
s 2 − 400 − 82s
s 3 + 10s 2 + 50s .
(20 marks)
QUESTION 6 (60 marks)
The height of a robot arm (in cm) at time t (in seconds) is given by
h = 2cos
? 3πt
5
?
+ 4cos
? 2πt
5
?
− 4cos
? πt
5
?
+ 5.
(a) Show h can be rewritten as h = 8cos 3
? πt
5
?
+ 8cos 2
? πt
5
?
− 10cos
? πt
5
?
+ 1. (19 marks)
(b) Using (a) find algebraically all times , t ≥ 0, (in exact form i.e. in terms of fractions) when the
height of the robot arm is 7 cm i.e. h = 7.
Hint: periodic solutions can be represented in the form t = a+nT where a is one solution, T is
the period, and n is an integer n = 0,1,... (33 marks)
(c) What are the first four times when the height of the robot arm is 7 cm?
Give your answers exactly (as a fraction) and approximately as a decimal (rounded to 2 decimal
places). (8 marks)
QUESTION 7 (40 marks)
The speed, V (km/h), of a car at time t (seconds) is given by
V = f(t) =
80 − 10sin
? π(t − 60)
120
?
2 − sin
? π(t − 60)
120
?
where 0 ≤ t ≤ 120.
(a) We want to know when the car will reach a certain speed. Solve for the time t in terms of the
speed V , and hence find the formula for the inverse function f −1 (t). (20 marks)
(b) Write down the domain of f −1 (t). (5 marks)
(c) Plot f(t) from t = 0 to t = 120 and then plot f −1 (t) on the same axes but only for the values
of t in the domain of the inverse function found in (b). (6 marks)
(d) Describe the geometric relationship that you can see when you consider both f(t) and f −1 (t).
(3 marks)
(e) When will the speed of the car be equal to 34 km/h?
Use the inverse function you found in part (a) to help you answer this. (6 marks)
Note: You are encouraged to use Matlab or Scilab for your plots but you may use other software,
including online graphing software like Desmos or GeoGebra, if you wish.
Note that the Matlab commands for sint and sin −1 t are sin(t) and asin(t), where t is in radians.
First define values of t from 0 to 120 for your plot of f.
The command
t1=[0:0.01:120];
gives values from 0 to 120, in steps of 0.01.
Then find the V -values for these t-values using
y1=(80-10*sin(pi*(t1-60)/120))./(2-sin(pi*(t1-60)/120));
Then plot f using the command
plot(t1,y1).
Use the command axis equal to get equal scales on the axes.
To add the plot of f −1 , define new t’s for its domain. That is with t2=[...] and then calculate the
y-values.
For instance for the function 7sin −1
? 5t + 1
3t − 2
?
− 1 we would use the command
y2 = 7*asin((5*t2+1)./(3*t2-2))-1;
We can then plot both functions on the same graph using the command
plot(t1,y1,t2,y2).
Include a plot style to distinguish the graphs, if you wish.
To add an additional plot of y = t to show the geometry, use the same process: define the t’s you want
for the plot first (t3=[...]), calculate the y’s, then plot by including t3,y3 after y2 to the previous
command plot(t1,y1,t2,y2,t3,y3).