Instructions:
Attempt ALL questions.
Write down your answers in the space provided.
Submit your Mathematica script as well as this handout (with your answers)
on Canvas. Instructions will be given later.
Task weight: 6%.
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 INFORMATION
REGARDING 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 as
stated 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 misconducts
at Insearch as stated in the Insearch Academic Handbook:
“…While studying at UTS:INSEARCH you are expected to
maintain high standards of academic honesty and integrity.
You will be penalised if you seek to gain unfair advantage by
copying another student’s work, or in any way misleading a
lecturer or tutor about your knowledge, ability, or the amount
of original work you have done, or if you assist other students
to do so…”
EMTH001 Homework 2 3
QUESTION 1 5 marks
3D Graphs and Contour Maps
You are asked to produce the graph shown below:
Hints.
• You should practice on section 4.5 “Combining 3-D graph and contour
map” 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
+
x
5
− y5´e−x2−y2
−
1
3
e−(x+1)2−y2
.
• Use 20 contour curves.
• The ticks on the graph above are very helpful.
Semester 1/2020
EMTH001 Homework 2 4
QUESTION 2 5 marks
The Method of Least Squares
You are assigned a set of 100 data points. Depending on your lab group, find
your 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 polynomial
f (x) = a3x3
+a2x2
+a1x+a0
that 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 in
part (a).
(c) Repeat part (b) with a quartic polynomial
f (x) = a4x4
+a3x3
+a2x2
+a1x+a0.
(d) Repeat part (b) with a quintic polynomial
f (x) = a5x5
+a4x4
+a3x3
+a2x2
+a1x+a0.
(e) Repeat part (b) with a sextic polynomial
f (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 1/2020
EMTH001 Homework 2 5
QUESTION 3 20 marks
DOUBLE INTEGRALS
You are assigned an individual region R. Depending on your lab group, find
your 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 passing
through A, C and D.
Notes. AC is a horizontal line.
TASKS
(a) Find the equation of lines AB, BC and the parabola ADC. Write them
on the table below:
Line AB : y= f1(x) = AAAAAAAAAAAAA
Line BC : y= f2(x) = AAAAAAAAAAAAA
Parabola ADC : y= f3(x) = AAAAAAAAAAAAA
Hints.
(1) The equation of the straight line passing through two given points
(x1, y1) and (x2, y2):
y =
y2− y1
x2− 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
a
b
c
=
x2
1 x1 1
x2
2 x2 1
x2
3 x3 1
−1
·
y1
y2
y3
.
Semester 1/2020
EMTH001 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:
a
Semester 1/2020
EMTH001 Homework 2 7
(iii) Describe R2 by using vertical cross-sections:
a
(iv) Describe R2 by using horizontal cross-sections (harder):
a
Semester 1/2020
EMTH001 Homework 2 8
(c) Using Mathematica, give the exact value of the double integral
Ï
R ¡64− x2
− y2¢ dA
by 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 simple
region.
(iii) R1 as a vertically simple region and R2 as a horizontally simple
region.
(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 and
Fbelow(x) for the curve MN below the x-axis. These two functions
should be defined by cases (i.e. piecewise functions).
Semester 1/2020
EMTH001 Homework 2 9
• To define a piecewise function in Mathematica, for example, to
define the function:
f (x) =
−3x−10, if x · −2
−x2, if −2 < x < 1
1
2(3x−5), if x ¸ 1
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