Patrícia Xufre | Manuel Araújo | João Fanha
MIDTERM | Fall 2023/24
ANSWERS (Multiple Choice):
1.
5.
2.
6.
3.1.
7.1.
3.2
7.2.
4.
8.
CALCULUS II
Name (optional):
Student Nr.:
Multiple Choice (10 points)
Exercise 1 (5.5 points)
Exercise 2 (4.5 points)
GRADE:
Time to completion: 120 minutes
GENERAL EXAMINATION RULES:




This exam consists of 10 Multiple Choice questions and 2 open questions. Answer the 1st part in its dedicated space on page 1.
This is a closed-book midterm.
There will be no individual clarifications during the midterm.
Read the test carefully and in the 2nd part justify all steps of your resolution.
MULTIPLE CHOICE QUESTIONS
1. (1 point) Consider 𝑓: ℝ → ℝ and such that 𝑓 (𝑥) = −
.
Find the upper bound for the difference: 𝑓(𝑥 + 1) − 𝑓(𝑥) given by the Mean Value Theorem:
A) 0.
B)
.
C) −
D) − (
.
)
.
2. (1 point) Consider 𝑓: 𝐷 ⊆ ℝ → ℝ a homogeneous function of degree 𝒂. Let 𝑔 be the function defined by:
𝑔(𝑥, 𝑦) = 𝑓 𝑥 𝒃 𝑦, 𝑥𝑦 𝒃 .
The function 𝑔 is:
A) not homogeneous.
B) homogeneous of degree 𝒂 + 𝒃 + 𝟏.
C) homogeneous of degree 𝒂𝒃 + 𝟏.
D) homogeneous of degree 𝒂𝒃 + 𝒂.
Midterm | Fall 2022/23
3. Consider 𝑓: 𝐷 ⊆ ℝ → ℝ a homogeneous function of degree 𝟐 and 𝐶 . Let 𝑔 be the function defined by:
𝑔(𝑥, 𝑦) = 𝑓(𝑥𝑦, 𝑥𝑦).
Also assume that 𝑓(1,1) = 1.
3.1. (1 point) The value of
(1,1) is:
A) 2.
B) 1.
C) −2.
D) −1.
3.2. (1 point) If
(𝑥, 𝑦) = 𝑥 𝑓′( , ) (𝑥𝑦, 𝑥𝑦) and 𝐻 (1,1) = 1 0 , then the value of
0 1
A) 6.
B) 4.
C) 3.
D) 2.
4. (1 point) Consider ℎ: 𝐷 ⊆ ℝ → ℝ defined by:
ℎ(𝑥, 𝑦) = ln(𝑥 + 𝑦 ).
The Taylor’s expansion of degree 2 which approximates ℎ in a neighbourhood of (0,1) is:
A) 2𝑥 + 𝑥 − (𝑦 − 1) .
B) 2𝑦 + 2𝑥 − 2(𝑦 − 1) .
C) 2𝑥 + 2(𝑦 − 1) + 𝑥 − (𝑦 − 1) .
D) 2(𝑦 − 1) + 𝑥 − (𝑦 − 1) .
5. (1 point) Consider 𝑓: 𝐷 ⊆ ℝ → ℝ defined by:
𝑓(𝑥, 𝑦) =
𝑥
+ ln(1 + 𝑦) − 𝑦.
1−𝑥
The point (0,0) is:
A) not a stationary point of 𝑓.
B) a stationary point of 𝑓, but is not an extremum point.
C) a local maximizer.
D) a local minimizer.
2/12
(1,1) is:
Midterm | Fall 2022/23
6. (1 point) Consider 𝑔: 𝐷 ⊆ ℝ → ℝ defined by:
𝑓(𝑥, 𝑦) = 𝑒 ( )
The point (0,0) is a stationary point of 𝑓. Which of the following is a singular direction of the Hessian matrix
𝐻 (0,0)?
A) (1,1).
B) (−1,1).
C) (0,1).
D) (1,0).
7.
Consider 𝑓: 𝐷 ⊆ ℝ → ℝ , a function that is 𝐶 and such that the MacLaurin expansions of degree 2 of each
coordinate functions of 𝑓 are, respectively:
𝑃 (𝑥, 𝑦) = 1 + 𝑦 + 𝑦 and 𝑃 (𝑥, 𝑦) = 𝑥 + 𝑥 .
7.1. (1 point) The value of 𝑓(0,0) is:
A) (0,0).
B) (0,1).
C) (1,1).
D) (1,0).
7.2. (1 point) What is necessarily false?
A) 𝑓 is locally invertible in the neighbourhood of (0,0).
0 −1
B) 𝐽 (1,0) =
.
−1 0
C) 𝑓 is 𝐶 .
0 −1
D) 𝐽 (0,0) =
.
−1 0
8.
(1 point) Suppose that 𝐹(𝑥, 𝑦, 𝑧) = 0 implicitly defines 𝑦 = 𝑓(𝑥, 𝑧) in the neighbourhood of (𝑥, 𝑦, 𝑧) = (0,1,0). If
(𝑥, 𝑧) = 𝑦𝑒 + 𝑥𝑧, what is the value of
(0,0)?
A) 0.
B) 1.
C) −1.
D) −2.
3/12
Midterm | Fall 2022/23
2nd PART
Exercise 1. Consider the following system of equations (𝑆):
𝑥 + 𝑧 + 𝑦𝑤 = 0
𝑥𝑧 + ln 𝑦 = 0
a. (1.5 points) Show that (𝑆) implicitly defines a function 𝑓, (𝑥, 𝑧) = 𝑓(𝑦, 𝑤), which turns each (𝑦, 𝑤) in a
neighbourhood of (1, −1) into a (𝑥, 𝑧) in a neighbourhood of (1,0).
b. (1.5 points) Compute 𝐽 (1, −1).
c. (1 point) Show that
(1, −1) = 0.
d. (1.5 points) Knowing that
(1, −1) = −2 and
lim
( , )→( ,
(1, −1) = −1, justify that:
𝑧(𝑦, 𝑤) + (𝑦 − 1) + (𝑦 − 1) + (𝑦 − 1)(𝑤 + 1)
= 0.
)
(𝑦 − 1) + (𝑤 + 1)
4/12
Midterm | Fall 2022/23
5/12
Midterm | Fall 2022/23
6/12
Midterm | Fall 2022/23
Exercise 2. John likes to be always updated about the culture news. In order to fulfill this objective, he spends
monthly 104€ in music (𝑀), books (𝐵) and cinema (𝐶). Each song can be bought at the price of 1€, each book
can be acquired by 16€ and each ticket for cinema is 8€. John wants to maximize his utility that is given by:
𝑈(𝑀, 𝐵, 𝐶) = √𝑀𝐵 + ln 𝐶.
a. (1.5 points) Formalize John’s problem as an optimization problem with equality constraints (i.e. ignore any
nonnegativity conditions about the variables), clearly stating its decision variables, objective function and
constraints.
b. Solve the problem by using the Lagrange Multipliers Method.
b.1. (1 point) Write the Lagrangian function and find its stationary point, (𝑀∗ , 𝐵∗ , 𝐶 ∗ ).
b.2. (1 point) Knowing that on point (𝑀∗ , 𝐵∗ , 𝐶 ∗ ):
 The 2, 2 entry and the 3,3 entry of 𝐻 are negative
 The 2,3 entry of 𝐻 is positive
 And that its leading principal minor of order 4 is negative,
classify the point found in b.1.)
c. (1 point) If the price of each cinema ticket decreases to 6€, what is approximately the value of John’s new
Maxima Utility?
7/12
Midterm | Fall 2022/23
8/12
Midterm | Fall 2022/23
9/12
Midterm | Fall 2022/23
DRAFT
10/12
Midterm | Fall 2022/23
11/12
Midterm | Fall 2022/23
12/12