# Online Practical Test 1

Course Code: BSC128
Course Name: Numerical Methods
Question Paper Setter: Liu Meifeng
Academic Session: 2023/04 Question Paper:

Total No. of Pages: 3 Time Allocated: 2 hours
Additional Materials: -
Apparatus Allowed: -

INSTRUCTIONS TO CANDIDATES

- This paper consists of Three questions. Please answer ALL questions.
- Read the above information carefully to ensure you have the correct and complete question paper.
- Please follow the requirement of each section and write down all the corresponding answers on the answer book provided.
- Please note that presentation of solutions (clarity, coherence, conciseness, and completeness) is important. Show your work and organize your solution. Answer without proper justification may receive less, or even zero credit.
- Communication between candidates in any means is forbidden. Answers must be entirely individual candidate’s independent effort. If you are found sharing your solutions with other candidates, or suspected of doing so, you would be penalized accordingly.

DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO.

(Student ID: Full Name: )

## Question 1 (15 marks)

A particle starts at rest on a smooth inclined plane whose angle 𝜃 is changing at a constant rate 𝑑𝜃 𝑑𝑡= 𝜔 < 0, see the figurative illustration below.

At the end of 𝑡 seconds, the position of the object is given by 𝑋(𝑡)= −𝑔 2𝜔ଶቆ𝑒ఠ௧−𝑒ିఠ௧ 2−sin𝜔𝑡ቇ, assuming that 𝑔 = 32.17 ft/sଶ. (a) Assume the value for 𝑡 is 1, express the position function in terms of the angle 𝜔, that is, find the expression 𝑋(𝜔) when 𝑡 = 1. [5 marks] (b) Go to the website GeGebra, set the values of 𝑔 as mentioned, use 𝑥 as the variable to replace 𝜔 and define the function 𝐹(𝑥) to represent the position function 𝑋(𝜔) obtained in (a). Show the graph of the function 𝐹(𝑥). Use the command “Derivative” in GeoGebra to evaluate the derivative of 𝐹(𝑥) and draw the graph 𝐹′(𝑥). [5 marks] (c) Draw a horizontal straight line 𝑦 = 𝑦, where 𝑦= 1.7, indicate the intercept of function 𝐹(𝑥) and the straight line. Convert the interception into radian measure, show your answer in three decimal places by rounding. [5 marks] Instructions: Share your GeoGebra link by clicking on the “Share” button, copy the link and paste it on “Student Answer Book”. Export Image and copy to clipboard, paste the image to “Student Answer Book”. Write the answers for Question 1(a) and Question 1 (c) on the “Student Answer Book”. Use “Print Preview” to download the commands lines in your GeoGebra as PDF files, attach the pdf file along with “Student Answer Book”.

## Question 2 (15 marks)

Continue with Question 1. (a) Go to website Octave Online , set the value of 𝑔 as mentioned in Question 1, the value of time 𝑡 to be the last digit of your school ID plus 1 divided by 10, and the value of 𝑦 to be the last two digits of your school ID plus 1 divided by 10. For example, if your school ID is CST1709123, then the value of 𝑡 should be 𝑡= 0.4 and the value for 𝑦 should be 2.4. [3 marks] (b) Create a function handler named myFunX to express the functions 𝑋(𝜔) as described in Question 1. [3 marks] (c) Suppose the particle mentioned in Question 1 has moved 𝑦 ft after 𝑡 second., then you will need to solve an equation in the form 𝑋(𝜔)=𝑦. Please define a function handler named myFun to express the equation 𝐹(𝜔) ≡𝑋(𝜔)−𝑦= 0, and use Secant method (Algorithm 2.4) to find the rate 𝜔 at which angle 𝜃 changes to within 10ିହ by choosing two appropriate initial guesses. [6 marks] (d) Express your answer in degree measure and compare it with Question 1(c), make your comment on Octave by using %. [3 marks]

Instructions: Save your script in Octave Online as Practical1_2_ID.m, download it and insert it on “Student Answer Book”. Take a screenshot on your code and results, paste the screenshot on the on “Student Answer Book”. Share your script as Octave Bucket, copy the link and paste it on “Student Answer Book”. Print your Octave script as PDF file and attach the pdf file along with “Student Answer Book”.

## Question 3 (20 marks)

A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the speed is in feet per second. Time 0 3 5 8 13 Distance 0 225 383 623 993 Speed 75 77 80 74 72

(a) Go to website Octave Online , create three arrays named T, Dist, Speed to store the given data: Time, Distance and Speed, respectively, find the length of the arrays. [4 marks] (b) Apply Neville’s method (Algorithm 3.1) to the given data by constructing a recursive table myQ1 to determine the position of the car when 𝑡= 9.8 seconds. [8 marks] (c) Use the Divided Difference method (Algorithm 3.3) by constructing a recursive table myQ2 to determine the position of the car when 𝑡= 9.8 second. [10 marks] (d) Compare the results obtained by (a) and (b), make your comment on Octave. [2 marks]

Instructions: Save your script in Octave Online as Practical1_3_ID.m, download it and insert it on “Student Answer Book”. Take a screenshot on your code and results, paste the screenshot on the on “Student Answer Book”. Share your script as Octave Bucket, copy the link and paste it on “Student Answer Book”. Print your Octave script as PDF file and attach the pdf file along with “Student Answer Book”.

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