Quiz 1 (Sample)

The University of Sydney
School of Mathematics and Statistics
Quiz 1 (Sample)
MATH1001: Differential Calculus
Semester 1, 2012
Lecturers: Holger Dullin, Robert Marangell, Di Warren and Zhou Zhang
Family name:
Other names:
SID:
Day:
Time:
Room:
Signature:
Please write your answers in the answer boxes. Each question is worth one mark. There
are 15 marks in all.
Please note:
(a) Working will not be marked. Marks will be awarded on the basis of answers only.
(b) Answers will only be marked if they are written in the correct answer box.
(c) Non-programmable, non-graphics calculators are permitted.
(d) Your tutor will provide you with working paper. No other paper is allowed.
(e) Working paper will be collected, but not marked.
(f) You have 40 minutes to complete the quiz.
(g) By signing, you have agreed not to talk about the quiz problems until the release of
the results.
Quiz (a)
c 2012 The University of Sydney
Copyright 1
1. Use interval(s) to express the set
{x ∈ R | |x + 2| 6 4 and |x| > 2}.
Answer:
2. Compute |(2 + i)(cos( π6 ) − i sin( π6 ))|.
Answer:
3. Write down the characterizing condition(s) Answer:
on x and y for the set
{z = x + iy ∈ C| |z| < |z + i|}.
4. Compute (i + 1)6 .
Answer:
5. Given i is a solution, find all the other
solutions for
z 3 − (2 + i)z 2 + (2 + 2i)z − 2i = 0.
Answer:
6. Find all solutions for ez = 1 + i
in the Cartesian form.
Answer:
7. −1 is not in the range of the function
f : C → C given by f (z) = z 2 + 1.
True or false.
Answer:
8. Find all solutions for z 3 =
form.
1
i
in the Cartesian Answer:
Answer:
9. Find the centre of the circle given by
x = 2 − sin t and y = cos t for t ∈ R.
2
10. What is the minimal value of z for any point Answer:
on the surface z = ex+y − 2 in the domain
D = {(x, y) ∈ R2 | x > 0 and y > 0}?
Answer:
11. Find fx for f (x, y) = x cos y · ex
at (x, y) = (1, 0).
12. Find the equation for the tangent plane of Answer:
the surface z = sin(x + y) + 1 at (0, 0, 1).
13. Find the limit
x2 − y 2
Answer:
along the
(x,y)→(0,0) x2 + y 2
lim
line y = 2x.
Answer:
14. Find the critical point of the function
f (x, y) = x2 + 3xy + y 2 .
15. (Choice Question) The function f (x, y) has
Answer:
a critical point where fxx = 2, fyy = 1 and
fxy = fyx = 1. This critical point is a:
(A) local minimiser; (B) local maximiser; (C)
saddle point.
3
Answers
Question 1: √
[−6, −2)
Question 2: 5
Question 3: y > − 12
Question 4: −8i
Question 5: 1 + i, 1 − i
Question 6: ln22 + ( π4 + 2kπ)i, k ∈ Z
Question 7: False
√
√
Question 8: 23 − 12 i, i, − 23 − 12 i
Question 9: (2, 0)
Question 10: −1
Question 11: 2e
Question 12: z = x + y + 1
Question 13: − 35
Question 14: (0, 0)
Question 15: A
4