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MATH1052 Assignment 2
● Write your answers clearly. Illegible assignments will not be marked.
● Show all your working. Correct answers without justi cation 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 levia the Assignment 1 Gradescope submission link on Blackboard > Assignments.
● Submit all applications for extensions via the my.UQ portal.
● Undertaking this assessment deems your commitment to UQ's academic integrity pledge as summarised in the following declaration:
I certify that I have completed this assessment in an honest, fair and trustworthy manner, that my submitted answers are entirely my own work, and I make this declaration 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.
● Marking:
✕ The maximum mark for the assignment is 50. ✕ 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 ne tune some aspects of your justi cation/calculations.
* Mark of 1: You have demonstrated a good understanding of the topic and techniques involved, with clear justi cation 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 ne tune some aspects of your justi cation/calculations.
* Mark of 2: You have demonstrated a good understanding of the topic and techniques involved, with clear justi cation 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 ne tune some aspects of your justi cation/calculations.
* Mark of 3: You have demonstrated a good understanding of the topic and techniques involved, with clear justi cation and well-executed calculations.
1. The equation of a quadratic curve is
2x2 + √3xy + y2 - 10 = 0.
(a) (2 marks) Use MATLAB to plot the graph of the quadratic curve. What does the curve look like?
Just because the curve looks like .... doesn't mean it is .... Seeing is not believing in maths! We will now PROVE that it is indeed a ... ! Let us begin with the general equation of a quadratic curve:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
We will eliminate the xy-term in the equation by a rotation of the coordinate axes. Such a cool idea but it will take some work so brace yourself!
(b) (3 marks) Suppose P = (x, y) is a point on the xy-plane. Now rotate the coordinate axes anticlockwise about the origin through an angle Q. Let the coordinates of P with respect to the new coordinate axes be (x, , y, ) (see gure below). Show that
x = x, cos Q - y, sin Q, y = x, sin Q + y, cos Q.
Useful tip: Recall the sum formula for cosine and sine. If you can't remember what these are, go to the FYLC. I've posted the formula on one of the walls in the room :). While you're there, say hello to the FYLC tutors - they can de nitely help you with this question.
(c) (2 marks) The equation Ax2 +Bxy+Cy2 +Dx+Ey+F = 0, B 
0, maybe transformed into a new quadratic equation A,x,2 + B,x,y, + C,y,2 + D,x, + E,y, + F, = 0. Determine an expression for B, in terms of A,B, C and Q.
(d) (2 marks) To eliminate the cross term in the new quadratic equation in (c), show that Q must satisfy
The angle Q gives the angle of rotation of the coordinate axes to produce a quadratic equation with no cross term.
(e) (1 mark) Determine Q for the quadratic equation 2x2 + √3xy + y2 - 10 = 0.
(f) (3 marks) Find an equation for 2x2 + √3xy + y2 - 10 = 0 in the x,y,-plane. Describe the curve.
2. The gure below shows a point P (x0 , y0 ) on the parabola y2 = 4px. The line L is tangent to the parabola at P. The parabola's focus lies at F (p, 0). The ray L, extending from P to the right is parallel to the x-axis. We show that light from F to P will be re ected out along L, by showing that β equals Q. This result establishes the re ective property of parabolas.
(a) (3 marks) Show that tan β = 2p/y0 .
(b) (1 mark) Show that tan φ = y0 / (x0 - p).
(c) (3 marks) Use the identity
to show that tan Q = 2p/y0 . Hence show that Q = β .
(d) (3 marks) You have just proved the re ective property of parabolas! This property is used in car headlights and satellite dishes. Write an amusing essay (maximum 150 words) on an interesting application of the re ective property of parabolas. You may include illustrations in your essay.
(e) (2 marks) The gure below shows a method for constructing a parabola. You will require a ruler and a string the length of the ruler. Draw a line on a piece of paper.
This is the directrix of the parabola. Choose a point F on the paper. This is the focus of the parabola. Pin one end of the string to F and the other end of the string to the upper end of the ruler. Hold the string taut against the ruler with a pencil, and slide the ruler along the directrix. As the ruler moves, the pencil will trace a parabola. Explain
why. Hint: recall that a parabola is the set of points in a plane equidistant from a given xed point and a given xed line in the plane.
(f) (Optional) Try the construction in (e) for fun! Special MATH1052 Prize for best parabola model based on this idea. Submit your model to Poh (Room 67-556) by Friday 10 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.
3. Let
(a) (3 marks) Sketch (by hand) the graph of z = f(x, y).
(b) (3 marks) Is f continuous at (0, 0)? Justify your answer using the de nition of continuity.
(c) (3 marks) Use the sketch in (a) to guess and at (0, 0). Check that your guess is correct for using the de nition of the partial derivative.
(d) (2 marks) What conclusion, if any, can you make about the existence of partial deriva- tives and continuity for a function f(x, y)?
4. Let's learn some vector calculus ahead of time! Let r(t) = x(t)i + y(t)j + z(t)k be the position vector of particle at time t. The velocity vector is given by v(t) = i + j + k.
An important fact about the velocity vector is that it is tangent to the path of motion. That's all you need to know for this problem! In preparation for a spectacular Taylor Swift concert at UQ, MATH1052 students at UQ constructed a surface given by the equation xz2 - yz + cos(xy) = 1.
(a) (1 mark) Show that (0, 0, 1) is on the surface. This is the point on the surface where Taylor Swift will land.
(b) (3 marks) Construct the equation of the platform tangent to the surface at (0, 0, 1). Hint: in page 171 of the workbook, you have the following geometrical property about gradient vectors in the plane: If f is di erentiable at (a, b), the gradient ▽f (a, b) is perpendicular to the contour line through (a, b).” Generalise this result to gradient vectors in space to help you construct the equation of Swift's platform.
(c) (3 marks) Taylor Swift will glide through UQ's exhibition hall along the path r(t) = (ln t)i + (tlnt)j + tk, t > 0. Show that the curve is tangent to the surface. Phew! That was a smooth landing!
5. A MATH1052 trophy is constructed by rotating the following region about the x-axis.
(a) (2 marks) Compute the volume of the trophy.
(b) (2 marks) Find the volume V (r, s) of a similar solid created by rotating a region with dimensions r and s instead of 3 and 2 respectively.
(c) (3 marks) Ooops!! The original numbers "2" and "3" were rounded o numbers. The actual quantities are o by up to 0.5 in either direction. Use linear approximation to estimate the maximum possible error in your answer to (a).