Due on Blackboard on Friday 8 September 2023 at 16:00
Question 1 (20 marks)
For each of the following sequences, determine whether it converges or diverges. If
the sequence converges, prove which number or vector is the limit of the sequence. If the
sequence diverges, carefully explain why.
(a)
{
3n+ 1n
}
n∈N
(b)
{(
3 + n2
) (
1
2
)n}
n∈N
(c)
{
n3+5n2+1
(n2+1)(2n+2)
}
n∈N
(d)
{
1√
n
· sin 3n, e−n
}
n∈N
Question 2 (15 marks)
Draw the following sets. Explain whether they are open, closed, both, or neither.
Define the set of interior points and boundary points. Justify whether the set is or is not
compact. Draw the set
(a) S =
{
(x, y) ∈ R2 : x+ |y| ≤ 2}.
(b) S =
{
(x, y) ∈ R2 : (x− 3)2 + (y − 3)2 < 9 ; y ≥ x}.
(c) S =
{
(x, y) ∈ R2 : x ≥ −|y| ; y ≤ sinx}.
Question 3 (20 marks)
(a) Prove that f : R2 → R, (x1, x2) 7→ ln
(√
(sinx1)2x32
)
is continuous. You may
use results from one-variable calculus and refer to theorems from our lecture slides
without proof.
(b) Let f : R2 → R \ {(0, 0)} : (x1, x2) 7→ x
2
1x
3
2
x31x
2
2
. Prove that every function g : R2 → R,
such that g(x1, x2) = f(x1, x2) for all (x1, x2) ∈ R2 \ {(0, 0)}, is discontinuous at
(0, 0).
1
ECON 2050 SEMESTER 2 2023
Question 4 (20 marks)
(a) Write down the formula for a linear function g with domain R2 and codomain R that
has the property that g(7, 2)− g(3, 4) = 8 and g(3, 5)− g(5, 2) = 4.
(b) Consider the function f : R2 → R : (x1, x2) 7→ (2,−3) · (x1, x2)−
√
2. How does
the value of f change if (x1, x2) changes
(i) from (1,−3) to (2, 1)?
(ii) from (4,−2) to (−1, 3)?
Question 5 (25 marks)
Let
f : R2 → R : (x, y) 7→ x4 + y2 + 2x2y − 1
Answer the following:
(a) Graph f(x, 0) against x in two dimensions and graph f(0, y) against y in two dimen-
sions.
(b) Draw the countour set f(x, y) = 3
(c) Calculate the directional derivative of f at the point (1, 1) in the direction of the
vector (−1, 1).
(d) At the point (1, 1), find the direction of steepest ascent. That is, find the vector v∗
with length 1 that causes f to increase most rapidly.
(e) Find the equation of the tangent plane at the point (1, 1) and comment on the rela-
tionship between the tangent plane and the slope of steepest ascent.