MATH1052 ASSIGNMENT 2 SEMESTER 2 2023
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MATH1052 ASSIGNMENT 2
SEMESTER 2 2023
Due: Fri 13 Oct 2023 at 5pm
❼ Write your answers clearly. Illegible assignments will not be marked.
❼ Show all your working. Correct answers without justification will not receive full marks.
❼ Wherever possible, answers should be given in exact form.
❼ This assignment is worth 7.5% of the total assessment for the course.
❼ Submit your assignment as a single pdf file via the Assignment 2 Gradescope submission link on Blackboard>>Assignments. Remember to assign your pages correctly.
❼ Submit all applications for extensions via the my.UQ portal.
❼ Marking: The maximum mark for the assignment is 30.
– 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.
1. (a) (3 marks) Consider the function f : R2 一 R defined by
Is f continuous everywhere? Justify your answer.
(b) Let g : R2 一 R be defined by
(i) (2 marks) For what value(s) of a, if any, is g continuous at (0, 0)?
(ii) (2 marks) For these value(s) of a, show that gx (0, y) = -2y for any y e R and gg (x,0) = 2x for any x e R.
2. Consider the vectors u = (-1, -2), v = (^3, -1), w = (-3, 2) and a smooth function f : R2 一 R such that fu (2, 1) = 0, fv (2, 1) = -1.
(a) (3 marks) Compute Δf(2, 1).
(b) (1 mark) Compute fw (2, 1).
3. Let C be the curve given by the points (x,y) e R2 satisfying
y3 + y2 cosx = x3 - 3x.
(a) (3 marks) Find d(d)x(g) on C. Are there any points on C for which dx(dg) does not exist?
Explain your answer.
(b) (2 marks) Find the equation for the tangent line at the point (0, -1).
4. Let w = (x2 + y2 )xy, x = rcosθ and y = rsinθ .
(a) (3 marks) Use the chain rule to find . Simplify your answer. (b) (1 mark) Find at the point (r,θ) = (2, .
5. Let f(x,y) = (x + y)e-2xg .
(a) (2 marks) Find the equation of the plane tangent to the surface z = f(x,y) at the point (1, 0).
(b) (2 marks) Find critical points.
(c) (3 marks) Find the linear and quadratic approximation functions of f at the critical point in the third quadrant. As a result, the linear approximation will be z = C, with C being a constant. Why does this occur?
(d) (3 marks) Use the quadratic approximation function found in part (c) to investi- gate the nature of the critical point located in the third quadrant. Hint: Complete the square. What quadratic surface does Q(x,y) resemble?
2023-10-28