This document provides instructions for a computing engineering laboratory assignment involving root finding methods in MATLAB. It consists of 9 tasks testing skills with false position, Newton-Raphson, secant, and modified secant methods. Students are asked to write M-files to locate roots of equations describing physical systems like bungee jumping, chemical reactions, and tank volumes. They must analyze solutions, compare methods, and debug provided code. The tasks involve both numerical techniques and plotting/graphical analysis skills relevant to engineering applications.

1. ENG1060 – Computing for Engineers
Laboratory 7 Page 1 of 9
This laboratory comprises 2% of your final grade.
During your lab session, you will be assessed on your programming
style as well as the results produced by your programs.
Save your work in M-Files called lab7t1.m, lab7t2.m etc
The questions are designed to test your recollection of the lecture material
up to and including lecture 14.
Note: Some of the functions presented would be new to you. You can use
the MATLAB help system to learn more about them.
Task 1
Write an M-file that uses the false position method to determine the mass of the
bungee jumper with a drag coefficient (cd) of 0.25kg/m to have a velocity of 36m/s
after 4s of free fall. (Note: acceleration due to Earth’s gravity is 9.81m/s2). Solve
the equation given below for m.
)(tanh)( tvt
m
gc
c
gm
mf d
d









Your M-file should prompt the user to enter the lower limit and upper limit and
check if the initial guesses bracket a root. If not, your program should prompt the
user for new guesses until the user provides values that do bracket the root.
Your M-file should print the iteration number, xl, xu, xr and f(xr) for each iteration.
Test your M-file for a precision of 0.5% using various initial guesses.
HINT: Substitute your root back into f(m) to check your answer
HINT: You can use graphical method to obtain initial guesses
HINT: You can reuse code provided on Moodle
Faculty of Engineering
Semester 1 - 2016
ENG1060 Computing for Engineers
Laboratory 7

2. ENG1060 – Computing for Engineers
Laboratory 7 Page 2 of 9
Task 2
Consider the following function:
10002008)( 23
 xxxxf
Locate the maximum of f(x) for x [-10, 10]. Do this by finding the root of the
derivative of this function. Use the Newton Raphson method to perform root finding.
Select the initial guess yourself after looking at f(x) graphically. Your solution
should achieve a precision of 0.01%. Plot f’(x) and the root on the same figure
Task 3
In a chemical engineering process, water vapor (H2O) is heated to sufficiently high
temperatures that a significant portion of the water dissociates, or splits apart, to
form oxygen (O2) and hydrogen (H2):
H2O« H2 +
1
2
O2
If it is assumed that this is the only reaction involved, the mole fraction, x, of H2O
that dissociates can be represented by:
Where K is the reaction’s equilibrium constant and pt is the total pressure of the
mixture.
If pt =3.5 atm and K = 0.04, write two M-files that uses
a) Bisection method to determine the value of x that satisfies the reaction
equation.
Your M-file should prompt the user to enter the lower limit and upper limit.
If the initial guesses do not bracket a root, your program should keep
prompting the user for new guesses until the user provides values that do
bracket the root.
Your M-file should print the iteration number, xl, xu, xr and f (xr) for each
iteration.
Test your M-file for a precision of 0.1%.
HINT: You can modify the root finding codes provided on Moodle
HINT: You can use graphical method to obtain initial guesses.

3. ENG1060 – Computing for Engineers
Laboratory 7 Page 3 of 9
b) Modified Secant method to determine the value of x that satisfies the
reaction equation.
Your M-file should prompt the user to enter the initial guess and
perturbation factor and print the root to 4 decimal places.
Test your M-file for a precision of 0.1% using various initial guesses.
HINT: You can modify the root finding codes provided on Moodle.
Task 4
You are designing a spherical tank (see figure below) of radius R = 3m, to hold
water for a remote village.
The volume of liquid it can hold can be computer as
where R is the tank radius, h is the depth of water and V is the volume of water
(a) Write an M-file to plot the function and then prompt the user to enter the
initial guess.
(b) In the same M-file, use Modified Secant function file to determine the
depth that the tank must be filled to, so that it holds 30m3 of water. Use a
precision of 0.001% and a perturbation of 0.01.
(c) Modify the modified secant function so that it outputs both the root and the
number of iterations. In the same M-file, print the number of iterations,
approximated root and fzero() root in Command Window.

4. ENG1060 – Computing for Engineers
Laboratory 7 Page 4 of 9
Task 5
A four-bar linkage system is shown above. The first link, a, is an input link (crank) of
length 1. The second link, b, is a coupler link of length 3. The third link, c, is an output
link of length 4. The forth link, d, is the fixed link (ground) of length 5. All lengths are
provided in metres.
The angular position of the output link () of a four-bar linkage corresponding to the
angular position of the input link () can be computed using the Freudenstein’s
equation:

The following parameters must be used for root finding:
x1120°, xu 165°, xi 120°, xi-1 110°, 0.01, precision of 0.05%.
a = 1; b = 3; c = 4; d = 5;
(a) Write an M-file to find the value of for  30°using the Newton-Raphson
method, Secant method and the Modified Secant method.
(Note: You may use MATLAB’s built-in function fzero() to check your
answer.)
(b) Plot the number of iteration versus the absolute percentage error. Use different
data marker and color for different method. Make certain that proper legend is
used.
(c) Compare the efficiencies of the Newton-Raphson method, Secant method and
the Modified Secant method. Which method requires the least iterations to
reach the required precision? Print your explanation in the Command Window
using short sentences. Use a new line for each sentence.

5. ENG1060 – Computing for Engineers
Laboratory 7 Page 5 of 9
Task 6
Many fields of engineering require accurate population estimates. For example,
transport engineers might find it necessary to determine separately the population
growth trends of a city and an adjacent suburb. Given that the population of a city is
declining with time according to
while the population of an adjacent suburb is growing according to
Using the following parameters:
Pc,end=75,000; kc=0.045/yr; Pc,start=100,000; Ps,end=300,000; Ps,start=10,000;
ks=0.08/yr;
Write an M-file to perform the following tasks:
(a) In order to determine the time when the suburb is 20% larger than the city, write
an anonymous function that define the root finding problem in the form of f(x)=0.
Plot a graph of f(x) where x covers 100 years period from the 1st year to the
100th year.
(b) Prompt the user to input the initial guess based on the graph plotted in (a).
(c) Use the False Position method to determine how many years later the suburb
is 20% larger than the city. Use a precision of 0.001%
(d) Assuming that the starting population of the city and suburb (Pc,start & Ps,start)
were recorded in year 2010, calculate which year (round to the nearest year)
the suburb is 20% larger than the city. Print a short sentence to show your
answer.
(e) Calculate the population of the city, Pc(t), and the suburb, Ps(t), when the
suburb is 20% larger than the city based on the answer in (c). Print another
short sentence to show these results.

6. ENG1060 – Computing for Engineers
Laboratory 7 Page 6 of 9
Optional (Ungraded Questions)
Task 7
The figure below shows a uniform beam of length, L, subject to a linearly increasing
distributed load, wo.
The equation for the resulting elastic curve is
where y is the deflection, wo is the distributed load, E is the Modulus of Elasticity
the material, I is the moment of inertia and /L is the length of the beam.
(a) Write an M-file that prompts the user to choose between using the False
Position Method or Newton-Raphson method. Both methods should be
prepared as user defined functions. (Note: At the end of the program, there
is no need to ask user if they want to try again)
(b) In the same M-file, write MATLAB program to determine the point of
maximum deflection to a precision of 1e-10 and the value of the maximum
deflection by finding the root of the derivative of the function (i.e find x where
dy/dx 0 and substitute x in the equation above to calculate y) given L =
0.6m, E = 50,000 kN/cm2, I = 30,000cm4, wo = 2.5kN/cm.
(c) Which method is provides a better solution? Justify your answer in less than
30 words. In the same M-file, print your answer in the Command Window
using fprintf() regardless of the choice of the user input.

7. ENG1060 – Computing for Engineers
Laboratory 7 Page 7 of 9
Note: Be careful with units
Task 8
A colleague new to MATLAB gave you the M-File lab7t8_bugs.m. Correct the
bugs in the file so that it performs modified secant correctly. Answer the questions
in bold as comments in your M-File.
(a) When I attempt to run the M-file, I got the following message:
??? Error: File: lab6t2_bugs.m Line: 17 Column: 34
Unbalanced or unexpected parenthesis or bracket.
At which line does this error occur? How do you correct this error?
(b) After correcting the error in part (a), I attempt to run the M-file again. This time
MATLAB complains about something else:
??? Undefined function or variable 'fxpx'.
Error in ==> lab6t2_bugs at 28
xi = xi-pert*xi*fxi/(fxpx-fxi); % Calculate new estimate
At which line does the real error occur? Why does the error occur? How
do you eliminate this error?
(c) After correcting the errors in part (b), I make my third attempt at running this M-
file. It gives me no error and printed the answer. 
The root of the equation is -Inf
Wait! The answer is wrong!!! 
Which line in the M-file is incorrect? What changes are required to obtain
correct answer? What was the mistake?

8. ENG1060 – Computing for Engineers
Laboratory 7 Page 8 of 9
Task 9
Aerospace engineers sometimes compute the trajectories of projectiles such as
rockets. A related problem deals with the trajectory of a thrown ball. The trajectory
of a ball is defined by the (x, y) coordinates as shown in the figure below.
The trajectory can be modeled as:
where is the angle in degrees, vo is the initial velocity, x is the distance from the
thrower to the catcher, y0 is the thrower’s elevation and y is the catcher’s elevation,
g is gravity (g = 9.81ms-2)
(a) Write a function that uses the secant method to calculate the root and
number of iterations to get the root. Your function should accept the function,
f, initial guesses xi, xi_1 and the precision as inputs.
function [root, iter] = secant (f,xi,xi_1,precision)
(b) Write an m-file that uses the function from part (a) to find the appropriate
initial throw angle and the number of iterations it takes to find the angle
given that vo = 20m/s ; x = 35m ; y0 = 2m; y = 1m and precision = 0.1%
Note: Your M-file should plot the trajectory path for 0 ≤ ≤ 60 and then prompt
the user to enter the initial guesses (they should be able to guess the initial values
after seeing the graph)

9. ENG1060 – Computing for Engineers
Laboratory 7 Page 9 of 9
Useful Information
When you demonstrate your work in this laboratory session, we will be looking out for
the following:
a) Does the code work accurately and work without intervention?
b) Are there indentations and comments explaining the functionality of the
code/function?
Hint:
% A comment field/line in MATLAB is always preceded
% with a ‘%’ symbol, after which it will turn green –
% indicating it is not a comment.
c) Are you able to create function which takes/outputs variables?
d) Are you able to perform root finding methods?
e) Are you able to justify which root finding method is better?
f) Are your functions/ iterations/ Program control statements properly indented?
After you have completed the demo. Please log on to Moodle and upload your files
there. You will only receive your demo marks if the corresponding files have been
uploaded.