MATH 2150
Real and Complex Analysis
Semester 1, 2021School of Science
MATH2150 Assignment 2 Due: 17:30pm, 2nd September, 2021
questions week
1 ～ 4 4
5 ～ 8 5
9 ～ 10 6
(5 + 5 + 5 + 5 + 4 + 5 + 6 + 3 (bonus)+ 5 + 5 (bonus) = 48 marks)
Your final mark for Assign 1 = min{40, total mark (inc. bonus)}
1. (5 points) Let f : D ? R→ R be given as the following rational function
f(x) =
2x4 ? 3x2 + x? 1
x2 + x? 6 .
Identify the largest possible domain of f .
2. (5 points) Let f : [?1, 1] → R be defined as f(x) = ?x? (i.e. the ceiling function, which gives
the smallest integer greater than or equal to x). Sketch the graph of this function, identify its
range and the points at which it is discontinuous on [?1, 1].
Note that there is no need for a formal proof here, just a brief explanation is enough.
3. (5 points) Prove that the function f(x, y) given below is discontinuous at (0, 0).
f(x, y) =
{
x
y , y ?= 0,
0, y = 0,
4. (5 points) Prove that if f : D ? Rn → R is continuous at xˉ ∈ D, then the function
h(x) = c · f(x),
where c is an arbitrary constant in R, is also continuous at xˉ.
5. (4 points) Calculate the derivative using the definition directly for
f(x) =
√
x for x > 0.
6. (5 points) Let f(x) = 3 lnx, g(x) = ln 2 · (x2?1). Prove that there exists a point c ∈ (1, 2) such
that f ′(c) = g′(c).
Hint: use Rolle’s theorems of differential calculus from Week 5 lecture slides
7. (6 points) Using L’Hospital’s rule, find the limit
lim
x→0
ln cosx
ln cos 3x
.
Hint: you may also use the fact that limx→0 u(x) · v(x) = limx→0 u(x) · limx→0 v(x), whenever
the individual limits on the right hand side exist.
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8. (optional, 2 bonus marks) Let
F (x, y, z) = z3 ? xz + y.
Verify that around the point (xˉ, yˉ, zˉ) = (2, 1, 1) the equation F (x, y, z) = 0 defines an implicit
function z = f(x, y) and conclude that by the implicit function theorem this function f is
differentiable at (2, 1). Find both partial derivatives of f at (2, 1).
9. (5 points) Using properties of Riemann integral show that if f(x) is an integrable odd function
on [?a, a], then ∫ a?a f(x)dx = 0.
10. (optional, 3 bonus marks) Compute the integral
K =
∫∫
P
(√
x+
√
y
)3
dxdy,
where
P =
{
(x, y)
∣∣x ≥ 0, y ≥ 0, √x+√y ≤ 1} .
Hint: let x = r cos4 t, y = r sin4 t (0 ≤ r ≤ 1, 0 ≤ t ≤ pi/2).
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