MATH1052 Assignment 2 Due: Fri 12 May, 5 pm
• Write your answers clearly. Illegible assignments will not be marked.
• Show all your working. Correct answers without justification will not receive full marks.
• Wherever possible, answers should be given in exact form.
• This assignment is worth 7.5% of the total assessment for the course.
• Submit your assignment as a single pdf file via the Assignment 2 submission link on Black-
board.
• Submit all applications for extensions via the my.UQ portal.
• Marking:
– The maximum mark for the assignment is 40.
– Marking Scheme for questions worth 1 mark:
∗ Mark of 0: You have not submitted a relevant answer, or you have no strategy
present in your submission.
∗ Mark of 1/2: You have the right approach, but need to fine tune some aspects of
your justification/calculations.
∗ Mark of 1: You have demonstrated a good understanding of the topic and techniques
involved, with clear justification and well-executed calculations.
– Marking Scheme for questions worth 2 marks:
∗ Mark of 0: You have not submitted a relevant answer, or you have no strategy
present in your submission.
∗ Mark of 1: You have the right approach, but need to fine tune some aspects of your
justification/calculations.
∗ Mark of 2: You have demonstrated a good understanding of the topic and techniques
involved, with clear justification and well-executed calculations.
– Marking Scheme for questions worth 3 marks:
∗ Mark of 0: You have not submitted a relevant answer, or you have no strategy
present in your submission.
∗ Mark of 1: Your submission has some relevance, but does not demonstrate deep
understanding or sound mathematical technique. This topic needs more attention!
∗ Mark of 2: You have the right approach, but need to fine tune some aspects of your
justification/calculations.
∗ Mark of 3: You have demonstrated a good understanding of the topic and techniques
involved, with clear justification and well-executed calculations.
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MATH1052 Assignment 2 Due: Fri 12 May, 5 pm
Attach this page to the front of your submission. Remember to sign the declaration.
I hereby state that the work contained in this assignment has not previously been
submitted for assessment, either in whole or in part, by either myself or any other
student at either The University of Queensland or at any other tertiary institution
except where explicitly acknowledged. To the best of my knowledge and belief, the
assignment contains no material that has been previously published or written by
another person except where due reference is made. I make this Statement in full
knowledge of an understanding that, should it be found to be false, I will be subject
to disciplinary action under Student Integrity and Misconduct Policy 3.60.04 of the
University of Queensland. The University of Queensland ’s policy on plagiarism can
be found at
http://ppl.app.uq.edu.au/content/3.60.04-student-integrity-and-misconduct
(Reference 3.60.04).
Name .......................................................................Student ID .........................
Signed .......................................................................Date .................................
Question Mark
1 /3
2 /15
3 /6
4 /3
5 /3
6 /10
Total /40
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MATH1052 Assignment 2 Due: Fri 12 May, 5 pm
1. (3 marks) The figure below shows a method for constructing a hyperbola. A corner of a
ruler is pinned to a fixed point F1 and the ruler is free to rotate about that point. A piece
of string whose length is less than that of the ruler is tacked to a point F2 and to the free
corner Q of the ruler on the same edge as F1. A pencil holds the string taut against the top
edge of the ruler as the ruler rotates about the point F1. Show that the pencil traces an arc
of a hyperbola with foci F1 and F2.
Hint: recall that a hyperbola may be defined as the set of points in a plane, the difference of
whose distances from two fixed focal points is a constant (Workbook p.121).
Try this construction for fun!
BONUS! Special MATH1052 Prize for best hyperbola model based on this idea. Submit your
model to Poh (Room 67-556) by Friday 12 May, 5 pm. If Poh is not in her room, leave the
model on the table located outside her room. Please write your name and Student ID on the
model.
2. Let P = (x1, y1) be a point on the ellipse
x2
a2
+
y2
b2
= 1.
Let F1 and F2 be the foci of the ellipse. Let L be the tangent line to the ellipse at P = (x1, y1).
Let α and β be the angles between the lines PF1, PF2 and L as shown in the figure.
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MATH1052 Assignment 2 Due: Fri 12 May, 5 pm
(a) (3 marks) Prove the following useful result: If two lines L1 and L2 intersect at an angle
θ, then
tan θ =
m2 −m1
1 +m1m2
where m1 and m2 are the slopes of L1 and L2 respectively.
Hint: tan(x− y) = tanx− tan y
1 + tan x tan y
.
(b) (3 marks) Determine the slope of L in terms of a, b, x1 and y1.
(c) (3 marks) Show that tanα =
b2
cy1
. Hint: recall that c2 = a2 − b2. Warning! The algebra
will get messy! Your job is to clean up the mess!
(d) (3 marks) Use (c) to show that α = β.
(e) (3 marks) You have just proved the reflection property of ellipses! This property explains
how whispering galleries, elliptical billiards and lithotripsy work. Write an amusing
essay (maximum 150 words) on an interesting application of the reflection property of
ellipses. You can include illustrations in your essay.
3. (a) (3 marks) Describe the curve represented by the parametric equations x = at, y =
bt, z = ct, where a, b, c are not all zero. Show that the value of
xyz
x2 + y4 + z4
approaches 0 as (x, y, z)→ (0, 0, 0) along the curve x = at, y = bt, z = ct.
(b) (3 marks) The curve in R3 represented by the parametric equations x = t2, y = t, z = t
is described in the figure below. You will learn more about parametric equations of
curves later in the course.
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MATH1052 Assignment 2 Due: Fri 12 May, 5 pm
Use part (a) to show that the limit
lim
(x,y,z)→(0,0,0)
xyz
x2 + y4 + z4
does not exist by letting (x, y, z)→ (0, 0, 0) along the curve x = t2, y = t, z = t.
4. (3 marks) Suppose you are making a rectangular box. You would like it to have a width
of 1 metre, a depth of 2 metres and a height of 4 metres. If there is an error of up to 10
centimetres in each of your length measurements, estimate the worst-case error in the volume
of the box. Compare your estimate to the actual worst-case error. You are most welcome to
use a calculator for this problem.
5. (3 marks) Let f : R2 → R be a function which has continuous partial derivatives. Consider
four points A = (2, 1), B = (4, 2), C = (3, 4) and D = (5, 3). The directional derivative of f
at A in the direction of the vector −→AB is 2 +√2, and the directional derivative at A in the
direction of −→AC is 3 +√2/2. Calculate the directional derivative of f at A in the direction
of the vector −−→AD.
6. Consider the function f(x, y) = x4 + 2y3 and the point P = (1, 1, 3).
(a) (2 marks) Write the equation for the tangent plane to the surface z = f(x, y) at the
point P .
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MATH1052 Assignment 2 Due: Fri 12 May, 5 pm
(b) (2 marks) Use part (a) to approximate f(1.5, 1.5).
(c) Use MATLAB to graph the following, all on the same graph with domain {1 ≤ x ≤
1.5, 1 ≤ y ≤ 1.5}.
i. (2 marks) the horizontal plane through P
ii. (2 marks) the tangent plane in part (a)
iii. (2 marks) f(x, y)
Print the graph in part (c), rotated so that the point (x, y) = (1.5, 1.5) is at the front and
(a, b) = (1, 1) is at the back.