CE142 MATHEMATICS FOR ENGINEERS
LAB TEST 1
AUTUMN TERM 2023
School of Computer Science and Electronic Engineering
University of Essex
Submission due on Friday 10th November 2023 (week 6) at 13:59:59
Submit electronically via FASER
Plagiarism
You are reminded that this work is for credit towards the composite mark in CE142 and that the work you
submit must therefore be your own. Any material you make use of, whether it is from textbooks, the web,
your own previous submissions or any other source, must be acknowledged in your report, and the extent
of the reference clearly indicated.
Please see your handbook for rules regarding the late submission of assignments.
Submission instructions and assessment:
 This lab test counts for 15% of the overall mark for the module.
 The lab test consists of seven (7) exercises that must be solved through mathematical working on
paper and suitable MATLAB code.
 The MATLAB code for all exercises (answering all questions therein) should be written and saved
in a MATLAB M-file (*.m) or Live Code File Format (*.mlx) created with MATLAB’s editor.
Clearly indicate (using comments) which exercise and question each piece of code corresponds to.
Use the following convention to name your MATLAB script: LT1_RegistrationNumber.m ( or .mlx)
 Your MATLAB script should run without producing any errors when called from the MATLAB
command prompt and should produce the required output (figures, printouts etc.)
 Where required, mathematical workings and justifications should be included in a single Word or
PDF report. Make sure you state clearly in your document which exercise and question each piece
of mathematical working corresponds to. State your registration number on the first page of your
report. Do not include MATLAB code or MATLAB figures in the report. Use the following
convention to name your Word or PDF file: LT1_RegistrationNumber.docx (.pdf)
 Use Word Equation Editor, Latex or any other editor of your choice that can export a PDF to write
mathematical equations. Handwritten scanned solutions will also be accepted. However, make sure
that the handwriting is legible and the scans are of good quality. Points will be deducted if your
answers are illegible.
 Provide adequate mathematical justification for all your answers. Points will be deducted for failing
to provide adequate mathematical reasoning even for answers that are otherwise correct.
 Please do not zip the MATLAB script and the report when uploading to FASER.
 This lab test is to be done individually. Every student should submit their own report and MATLAB
code. Automatic and manual plagiarism checks will be applied.
 Your mark (with written feedback) will be returned to you within four weeks in the term time.
Lab Test 1
Exercise 1
Q1. Consider the function () = 2 + 6. Calculate the function’s inverse −1()
and deliver your workings in the report. Then, write the MATLAB commands that
will produce the plot of −1(). Add labels to the axes and a legend to your figure.
For this and any subsequent plots, make sure your plot illustrates an appropriate
function support and codomain (including, where applicable, both negative and
positive values for the - and -axis).
[5 pts]
Q2. Consider additionally the function () = 3 − 5. Calculate the composite
functions (()) and (()) and deliver your workings in the report. Then, write
the MATLAB commands that will produce the plots of (()) and (()) on
the same figure. Use different colours and line styles for the two graphs, and add a
legend to your figure.
Hint: In order to plot in a new MATLAB figure without overwriting your previous
plots, use the MATLAB command ‘figure()’ before the code which creates the
desired plot. ‘figure()’ will create a new empty figure with increasing index. All
graphics will be forwarded to this new figure until the next call to ‘figure()’. Apply
this technique to all subsequent questions and exercises to avoid overwriting the
requested outputs of previous questions. In order to add plots to the current figure
without deleting existing plots use the MATLAB command ‘hold on’.
[5 pts]
Q3. Calculate the function rule of the linear function whose graph passes through
the point (7,4) and intersects the -axis at point with -coordinate equal to 3, and
deliver your workings in the report. Then, write the MATLAB commands that will
produce a figure with the graph of the function you previously found. Superimpose
on the same plot an asterisk (or any other marker of your choice) at points (7,4)
and (0,3). Verify that the graph of the linear function passes though these points.
[5 pts]
Exercise 2
Q1. Consider the function () = 2 + 2 − 3. Compute the factorised version of
() and deliver your workings in the report clearly showing the steps for obtaining
the factorised version. Then, write the MATLAB commands that will plot both the
graph of function (), and the graph of its factorized version in the same figure.
What do you observe? Use different colours and line styles for the two graphs, and
add a legend to your figure.
[5 pts]
Q2. Find the roots of () and deliver your workings in the report clearly showing
the steps for obtaining the roots. Then, write a MATLAB script that calculates the
roots of () and superimposes an asterisk for each root of the polynomial () on
[5 pts]
the graph of (). Does the output of your MATLAB code agree with your
workings?
Q3. You are told that the functions () and ℎ() derive from function () given
above, by changing only a single one of the polynomial coefficients of (), in each
case. State the function rules of () and ℎ() in the report (including an
appropriate justification), given that: a) () passes through the origin of the
Cartesian coordinate system and b) the parabola of ℎ() opens to the opposite
direction than that of (), and ℎ(1) = −2. Then, write the MATLAB commands
that plot the graphs of () , () and ℎ() on the same figure. Use different colours
and line styles for the three graphs, and add a legend to your figure.
[5 pts]
Q4. In addition to the function () of the previous questions, consider the function
() = + 1 and the equation () = (). Solve the equation analytically and
include your workings in the report. Then, write suitable MATLAB commands that
find the solution of this equation. Write additional MATLAB commands that create
a plot which illustrates graphically the solutions of the above equation. Superimpose
on this plot an asterisk for each solution of the equation. Does the output of your
MATLAB code agree with your workings?
[5 pts]
Exercise 3
Q1. Write MATLAB code printing one solution of the trigonometric equation
sin() = 0.5. Make sure the output is in degrees, not radians.
[5 pts]
Q2. Find the function rule of a waveform which oscillates between the values of
±10, has a period of 4 radians, has value 0 for = 0, and peaks to its negative
extremum before the positive one after = 0. Deliver your workings in the report.
Then, write MATLAB code to generate the plot of the waveform that you found.
Discuss in the report whether your plot agrees with the given specifications.
[5 pts]
Q3. Find all the solutions of the equation 1 2⁄ sin() + (√3 2⁄ ) cos() = √2 2⁄ in
the interval [−90°, 90°] and include your workings in the report clearly stating the
function properties/identities you used. Then, write MATLAB code that outputs
(prints) one solution of the above equation. Make sure that the solution is in
numerical format. Write additional MATLAB code that creates a plot which
illustrates graphically the solutions of the above equation. Clearly state in the script
(as a comment) or show on the plot where the solutions are located.
[5 pts]
Exercise 4
Q1. Solve the equation eln(
2−8) = ln(e) − ln() and deliver your workings in the
report. Then, write MATLAB code that prints the solution of the equation 10 =
log0.1(). Make sure that the printed solution is a real numerical value (not an
expression or a complex number).
[5 pts]
Hint: Use a graphical method first to determine an approximate solution of the
above equation and use this as the initial point for one of the MATLAB solvers.
Q2. Consider the generic form of the exponential function: () = . Select
suitable values for the parameters and for () to represent:
i) an exponential growth, where the output doubles itself at each step and the
graph of the function intersects the -axis at 0.5,
ii) exponential decay, where the output diminishes to one third of itself at each
step and the graph of the function intersects the -axis at 3.
Consider that the step is equal to 1. Deliver your workings in the report clearly
showing the steps for obtaining the values of the parameters and . Then plot the
graphs of the functions that you obtained with MATLAB. Verify that the plots
satisfy the above requirements.
[5 pts]
Exercise 5
Q1. In the RLC series circuit shown in Fig 1, with = 5, = 0.1 and =
0.5, you are told that the voltage () across the resistor was measured to be
() = 0.2cos(50 – /4). Consider Kirchoff’s laws known: () = () =
() = (), () = () + () + (), as well as that: () = (),
() = ()/ and () = ()/. Compute the voltage () across
the inductor and deliver your workings in the report. Then, use the Symbolic Math
Toolbox to write MATLAB code which computes and plots the voltage () across
the inductor . Does the output of your MATLAB code agree with your workings?
[6 pts]
Q2. You are told that the voltage () across the capacitor in the RLC circuit in
Fig. 1 is a sinusoidal function of the general form () = sin( + ). Compute
the voltage () (i.e, find the values of A, and ), and deliver your workings in
the report. Do not use integration in your solution. Write a MATLAB script which
plots the voltage () that you previously found. Make sure to use an appropriate
function support so that the waveform is clearly visible.
[7 pts]
Q3. Fig. 2 shows an LC oscillator circuit, where () = () = (). You are
told that the frequency of oscillation of () is = 1
2π√
⁄ . Consider that the
nominal values of capacitance and inductance are not known, but have a
[7 pts]
Figure 1: RLC circuit
constant sum of 10, that is + = 10. Compute the values of , , so that the
frequency of oscillations of () in the circuit is the minimum possible. Then use
the Symbolic Math Toolbox to write a MATLAB script which finds and outputs
the values of , that minimise the frequency of oscillations of () in the circuit.
Does the MATLAB output agree with your workings?
Exercise 6
Q1. Consider the function () = 3 + 32 – 2 − 1 . Find the point of the
graph of the function () where its tangent line has the smallest possible
gradient. Compute the function rule of the tangent line. Deliver your workings in
the report. Then, write a MATLAB script which plots () and its tangent line at
point , and annotates point with an asterisk.
[5 pts]
Exercise 7
Q1. Compute the following integrals and deliver your workings in the report. Then,
write the MATLAB commands that compute the same integrals using MATLAB
functions/toolboxes of your choice. Does the output of your MATLAB code agree
with your workings? Discuss any discrepancies in the report.
i) ∫(2 + 5)sin()
ii) ∫ e3+1
1
0
iii) ∫
3+2
1+2
1
−1
iv) ∫ sin2(2 + 4 + 10)e
[8 pts]
Q2. Compute the size of the yellow surface illustrated in the graph below and
deliver your workings in the report. Then, use MATLAB’s Symbolic Math Toolbox
to write a script, which calculates the size of the yellow surface without using any of
the results that you obtained analytically. Does the output of your MATLAB code
agree with your workings? Discuss any discrepancies in the report.
[7 pts]
Figure 2: LC oscillator circuit