# EMTH001 – Semester 1/2020 MATHEMATICAL MODELLING 2

by | May 22

Instructions:
Attempt ALL questions.
Submit your Mathematica script as well as this handout (with your answers)on Canvas. Instructions will be given later.
Penalties for late submission:☛ up to 3 days (till 00:00 Tuesday 26 May 2020): −2%;☛ from 3 to 7 days (till 00:00 Saturday 30 May 2020): −4%;☛ more than 7 days (after 00:00 Saturday 30 May 2020):no submission is accepted.
You must use the assigned data to complete your work. Otherwise,a zero mark will be given to the corresponding question.IMPORTANT INFORMATIONREGARDING LAB ASSESSMENT & ATTENDANCE
The total weight 20% of Lab Assessment is distributed as follows:Description Weight
Homework 1 (due in week 6) 26%
Homework 2 (due in week 10) 26%
Lab Test (week 11) 28%TOTAL 20%
Late submission of Homework will result in penalties being applied asstated on the cover page of the homework paper.
Missing the Lab Test will result in zero marks being recorded.
Students are reminded about severe penalties applied to academic misconductsat Insearch as stated in the Insearch Academic Handbook:“…While studying at UTS:INSEARCH you are expected tomaintain high standards of academic honesty and integrity.You will be penalised if you seek to gain unfair advantage bycopying another student’s work, or in any way misleading alecturer or tutor about your knowledge, ability, or the amountof original work you have done, or if you assist other studentsto do so…”EMTH001 Homework 2 3QUESTION 1 5 marks3D Graphs and Contour MapsYou are asked to produce the graph shown below:Hints.• You should practice on section 4.5 “Combining 3-D graph and contourmap” from the Lab Notes (page 53).• The function that describes the mountain surface is:f (x, y)= 3(1− x)2e−x2−(y+1)2−10³−x3+x5− y5´e−x2−y2−13e−(x+1)2−y2.• Use 20 contour curves.• The ticks on the graph above are very helpful.Semester 1/2020EMTH001 Homework 2 4QUESTION 2 5 marksThe Method of Least SquaresYou are assigned a set of 100 data points. Depending on your lab group, findyour data set in the Mathematica notebook:• LAB01 Tuesdays Question 2 DATA.nb, or• LAB02 Mondays Question 2 DATA.nb, or• LAB03 Mondays Question 2 DATA.nb.TASKS(a) Plot the given list of data points.(b) Use the method of least squares to find a cubic polynomialf (x) = a3x3+a2x2+a1x+a0that approximates the function y = g(x) described by the data points.Plot the cubic function y = f (x) together with the plot of data points inpart (a).(c) Repeat part (b) with a quartic polynomialf (x) = a4x4+a3x3+a2x2+a1x+a0.(d) Repeat part (b) with a quintic polynomialf (x) = a5x5+a4x4+a3x3+a2x2+a1x+a0.(e) Repeat part (b) with a sextic polynomialf (x) = a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0.(f) Which polynomial gives the best approximation to the given data points?Hint. Refer to section 6.2 “The Method of Least Squares” from the Lab Notes(page 78).Semester 1/2020EMTH001 Homework 2 5QUESTION 3 20 marksDOUBLE INTEGRALSYou are assigned an individual region R. Depending on your lab group, findyour region in the following PDF file:• LAB01 Tuesdays Question 3 DATA.pdf, or• LAB02 Mondays Question 3 DATA.pdf, or• LAB03 Mondays Question 3 DATA.pdf.The region R = R1[R2 is the union of the following two subregions:• subregion R1 which is the triangle ABC, and• subregion R2 which is the region bounded by the parabola passingthrough A, C and D.Notes. AC is a horizontal line.TASKS(a) Find the equation of lines AB, BC and the parabola ADC. Write themon the table below:Line AB : y= f1(x) = AAAAAAAAAAAAALine BC : y= f2(x) = AAAAAAAAAAAAAParabola ADC : y= f3(x) = AAAAAAAAAAAAAHints.(1) The equation of the straight line passing through two given points(x1, y1) and (x2, y2):y =y2− y1x2− x1 ¡x− x1¢+ y1 (Two-point form)(2) The equation of the parabola passing through three given points(x1, y1), (x2, y2) and (x3, y3):y = ax2+bx+ c where abc=x21 x1 1×22 x2 1×23 x3 1−1·y1y2y3.Semester 1/2020EMTH001 Homework 2 6(b) (This part must obviously completed by hands)(i) Describe R1 by using horizontal cross-sections:a(ii) Describe R1 by using vertical cross-sections:aSemester 1/2020EMTH001 Homework 2 7(iii) Describe R2 by using vertical cross-sections:a(iv) Describe R2 by using horizontal cross-sections (harder):aSemester 1/2020EMTH001 Homework 2 8(c) Using Mathematica, give the exact value of the double integralÏR ¡64− x2− y2¢ dAby using the following descriptions:(i) both R1 and R2 as vertically simple regions.(ii) R1 as a horizontally simple region and R2 as a vertically simpleregion.(iii) R1 as a vertically simple region and R2 as a horizontally simpleregion.(iv) both R1 and R2 as horizontally simple regions.(d) Plot the region R by using Mathematica. It should look like this:Note. This is my region. Please sketch yours.Hints.• Define functions Fabove(x) for the curve MN above the x-axis andFbelow(x) for the curve MN below the x-axis. These two functionsshould be defined by cases (i.e. piecewise functions).Semester 1/2020EMTH001 Homework 2 9• To define a piecewise function in Mathematica, for example, todefine the function:f (x) =−3x−10, if x · −2−x2, if −2 < x < 112(3x−5), if x ¸ 1
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