Lecturer: Approved by:

CODE: 4
..............................................................................................................

Semester/ Academic year 213 2021 - 2022

MIDTERM EXAM
Date 10 July 2022
UNIVERSITY OF TECHNOLOGY Course title Calculus 2
VNUHCM Course ID MT1005
FACULTY OF AS Duration 50 mins Question sheet code: 1764

-This is a closed book exam. Total available score: 10 points.

-At the beginning of the working time, you MUST fill in your full name and student ID on this question
sheet.

Student’s full name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Student ID: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 1:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 2:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I. Multiple choice (6 points, 60 minutes)

There are 20 questions.
Zx Each question is assigned 0.5 point. (L.O.1)
d p
Question 01. If f ( x ) = cos(t2 )dt, find A = f ( x ).
dx
0
cos x sin x cos x sin x
A =g p
Q. 1.A Let B Adefined
be the function = p by C A= p D A= p
3 x 3 x 2 x 2 x
Question 02. Find the approximation of3x
c such
 10  
3
−that
xyz 4 − 2xyz 1
g (x, y, z) = Z x ln .
(xf 2(t+ 4 3 z 6 )8 2
ex−(y−2) −z3
)dty= x++ 3x2 + 2x 3.
c

Evaluate gy (−1, 0, 0).


A c = 0.9717 0.8717
B c= 
C c = 0.6717 D c = 0.7717 
A −4245690 B −3753892 C 1845654 D −4194304

Question 03. If F ( x ) = x2 is an antiderivative of the function f ( x )e2x . Find the antiderivative of the function
Ef ( x−2344900
0 2x
)e . Z
A None of them B f 0 ( x )e2x dx = x2 + 
x+C 
Z Z
Q. 2. Choose a point in the domain of the function0 f 2x (x, y) = ln sin 2 1 2 from the
C f 0 ( x )e2x dx = x2 2x + C D f ( x )e dx = 2x2 + 2x + C x +y
following.
  ln
x
AQuestion 04. If
M (−0.3, F ( x ) is an antiderivative
−0.3) B M (0.1, of the function f ( x ) =
−0.3) · Calculate
C xM I = F (e) F (1).
(0.2, −0.2)
 
1 −0.3) 1
D AMI (0.2,
= B EI =M e (0.5, −0.2) C None of them D I=
e 2
2 2 2 Z
3. Let M
Q. Question 05. be a point
If
the onf the
functions and gcircle (x + 2) derivative
have continuous +y = 4.
onGiven
[0, 2] andthat M the
satisfies hascondition
polar coordinates
f 0 ( x ) g( x ) dx =
Z 2 r0 , − 3π (in the standard 0
4
Z 2polar
h coordinate
i0 system x = r cos φ, y = r sin φ). What
2, f ( x ) g0 ( x ) dx = 3, then calculate f ( x ) g( x ) dx.
0is the value of r 0 ? 0
√ √  
A

Stud.
3 2 B 2 2 C 4 D 2
√Fullname: Page 1/5 - Question sheet code 1254
E 2

Q. 4. Let (S) be the paraboloid z = 10 − (x − 1)2 − (y − 2)2 , and let M be the highest
point (relative to the plane Oxy) on (S). Let (P ) be the tangent plane to (S) at M .
Which point from the following is on the plane (P )?
  
A N (1, 2, 9) B N (−1, 2, 9) C N (−1, −2, 9)
 
D N (1, −2, −9) E N (3, 4, 10)

1 CODE: 1764
 Q. 5. Let u (x, y) be a function such that

ux (−1, 0) = −4, uy (−1, 0) = −2, u (−1, 0) = 3.

Consider the function F

 
(x, y) := cos (u (x, y)). 
Evaluate Fx (−1, 0).
A cos(4) B 4 sin(3) C 6 sin(3)
 
D 4 sin(3) + cos(4) E 6 sin(3) + cos(4)

Q. 6. Let M be a point different from the origin and on the intersection between the line
y − x = 0 and the parabola x − y 2 = 0. Which one from the following is the polar
coordinates of M (in the standard polar coordinate system x = r cos φ, y = r sin φ)?
  √   √
A M (1, π3 ) B M ( 2, π4 ) C M (1, − π4 ) D M ( 2, − π4 )

E M (1, π4 )
2 −y 2
Q. 7. Which number from the following is in the range of the function f (x, y) = ex ?
   
A −2.71 B 0 C −0.3 D 0.2

E −0.4

Q. 8. Let D = {(x, y) : x2 + y 2 ≤ 1, y ≤ −x}. In the polar coordinate system x =

r cos (φ) , y = r sin (φ), what is the range for (r, φ) corresponding to the domain
D?
Choose the correct statement from the following.

A 0 ≤ r ≤ 1, −π ≤ φ ≤ π

B 0 ≤ r ≤ 1, − π4 ≤ φ ≤ 3π
 4
C 0 ≤ r ≤ 1, − 5π ≤ φ ≤ − π4
 4
D 0 ≤ r ≤ 1, − π4 ≤ φ ≤ π
 4
E 0 ≤ r ≤ 1, 3π
4
≤φ≤ 3π
2

Q. 9. In R3 , let (P ) be the plane of equation 4x − 2y − 2z + 8 = 0. Which function in the

following whose graph is the plane (P )?

A f (x, y)

= 8x − 4y + 4
B f (x, y)

= 2x − 3y + 8
C f (x, y)

=x−y+2
D f (x, y)

= 2x − 2y + 1
E f (x, y) = 2x − y + 4
Q. 10. In R2 , what is the greatest distance from a point on the circle x2 + y 2 = 1 to the
point M (−1, 7)?
p √ p √ p √
A
p 51 + 10 2 B 50 + 53 C 50 + 18 2
 √ √
D 51 + 53 E 51
Q. 11. Find the maximum value of the function f (x, y) = 13 − x2 − y 2 for (x, y) in the
domain 
(x, y) : x4 + y 6 ≤ 1 .
 √    √
A 14 2 B 12 C 11 D 12 2

E 13

2 CODE: 1764
Q. 12. The points given in the following are represented in polar coordinates. Which point
is inside the circle x2 + y 2 = 4?





A 5, − π6 B 32 , π6 C 5, π3 D 3, π2


E 6, π4

Q. 13.How many saddle points


does the function f (x,y) = 2022 + 2021ye−x have?

A 0 B 1 C 2 D 3

E Infinitely many

Q. 14. Among the following directions, which one that the function f (x, y) =
1
x2 +y 4
sin (x2 + y 4 ) goes uphill most rapidly at M (−1, 1)?
   
A →−u (1, 0) B → −u (−2, 5) C →−u (1, −2) D →
−
u (3, −1)

→
−
E u (4, −6)
Q. 15. Evaluate the double integral ZZ
x2 dA
D

where D is the disk {(x, y) : x2 + y 2 ≤ 9}.

   
9π 81π 9π
A 9π B C D
 2 4 4
27π
E 4
p
Q. 16. Let (C)
p be the intersection between the surface z = 8 − x2 − y 2 and the cone
z = x2 + y 2 . The projection of (C) onto the plane Oxy is a circle. Evaluate its
radius.
   √
A 1 B 2 C 4 D 2 2
√
E 2 3
Q. 17. Evaluate the value of the function
 3 10  
3x − xyz 4 − 2xyz 1
g (x, y, z) = ln
(x2 + y 4 + z 6 )8 + 1
2
ex−(y−2) −z3

at M (x = 0, y = 2, z = 0).
   
A −2 ln(2) B 0 C 3 ln(2) D 2 ln(2)

E −3 ln(2)

Q. 18. In R3 , let (C) be the intersection between the cylinder x2 + y 2 = 5 and the plane
2x + 3y − 4z + 8 = 0. Which point from the following is on the curve (C)?
   
A M (1, 2, −1) B M (1, 2, 1) C M (−1, 2, 1) D M (1, −2, 1)

E M (−1, −2, 1)

Q. 19. Which function from the following has a local minimum at the point M (1, 0)?

A f (x, y)

= (x − 1)2 + y 2 + 3
B f (x, y)

= − (x − 1)2 − y 2 + 3
C f (x, y)

= (x − 1) y + 3
D f (x, y)

= (x − 1) y 2 + 3
E f (x, y) = (x − 1)2 y + 3

3 CODE: 1764
Q. 20. Evaluate the double integral ZZ
3dA
D

where D is the triangular region with vertices: A (−3, 0) , B (0, 5) and C (7, 0).
   
A 150 B 25 C 50 D 30

E 75

4 CODE: 1764
 HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

—————————————————
ANSWER KEYS
SUBJECT: CALCULUS 2
DURATION: 50 MINUTES
 NO MATERIALS ARE ALLOWED
CODE: 1764
    
Q. 1. D Q. 5. B Q. 9. E Q. 13. A Q. 17. B
    
Q. 2. D Q. 6. B Q. 10. A Q. 14. C Q. 18. D
    
Q. 3. B Q. 7. D Q. 11. E Q. 15. C Q. 19. A
    
Q. 4. E Q. 8. C Q. 12. B Q. 16. B Q. 20. E

1 CODE: 1764
 Lecturer: Approved by:
CODE: 5
..............................................................................................................

Semester/ Academic year 213 2021 - 2022

MIDTERM EXAM
Date 10 July 2022
UNIVERSITY OF TECHNOLOGY Course title Calculus 2
VNUHCM Course ID MT1005
FACULTY OF AS Duration 50 mins Question sheet code: 1765

-This is a closed book exam. Total available score: 10 points.

-At the beginning of the working time, you MUST fill in your full name and student ID on this question
sheet.

Student’s full name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Student ID: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 1:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 2:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I. Multiple choice (6 points, 60 minutes)

There are 20 questions.
Zx Each question is assigned 0.5 point. (L.O.1)
d p
Question 01. If f ( x ) = cos(t2 )dt, find A = f ( x ).
dx
0
cos x 2 2 sin x cos x sin x
A = Mpbe a point onB the
Q. 1.A Let A =circle
p (x + 2) +y C A= = 4.pGiven that M A = polar
D has p coordinates
3 3πx
3 x 2 x 2 x
r0 , − (in the standard polar coordinate system x = r cos φ, y = r sin φ). What
Question 02.4 Find the approximation of c such that
is the value of r0 ?
√  √ Zx √ 
A 3 2 B 2 f (t)dt = x3 + 3x2 +
C2x 2 3. 2 D 4
 c
E 2
A c = 0.9717 B c = 0.8717 C c = 0.6717 D c = 0.7717
Q. 2. Let D = {(x, y)2 : x2 + y 2 ≤ 1, y ≤ −x}. In the 2xpolar coordinate system x =
Question 03. If F ( x ) = x is an antiderivative of the function f ( x )e . Find the antiderivative of the function
f 0 ( x )re2x
cos
. (φ) , y = r sin (φ), what is the rangeZ for (r, φ) corresponding to the domain
D?
A None of them B f 0 ( x )e2x dx = x2 + x + C
Choose
Z the correct statement from the following.
Z

C 0 2x 2
f ( x )e dx = x 2x + C 0 2x 2
D f ( x )e dx = 2x + 2x + C
A 0 ≤ r ≤ 1, −π ≤ φ ≤ π

ln x

0 ≤ r04.
BQuestion ≤ 1,If F3π ≤ φ ≤ 3π
4( x ) is an antiderivative
2 of the function f ( x ) = · Calculate I = F (e) F (1).
x
π 3π
C A0 I≤= r1 ≤ 1, − 4 ≤ φ ≤ 1
 e
B I4 = e C None of them D I=
2
D 0 ≤ r ≤ 1, − 5π ≤ φ ≤ − π4
 4 Z 2
EQuestion
0 ≤ r05.
≤ 1, − π4functions
If the π g have continuous derivative on [0, 2] and satisfies the condition
≤ φ ≤f and
4 0
f 0 ( x ) g( x ) dx =
Z 2 Z 2h i0
0
Q. 2,
3. 0Let x ) dx
f ( x )gg (be the= 3, then calculate
function defined
0
( x ) g( x )
f by dx.

 10  
Stud. Fullname: 3x3 − xyz 4 − 2xyz Page 1/51- Question sheet code 1254
g (x, y, z) = ln .
(x2 + y 4 + z 6 )8
2
ex−(y−2) −z 3

Evaluate gy (−1, 0, 0).

   
A −4245690 B −2344900 C −3753892 D 1845654

E −4194304

1 CODE: 1765
 Q. 4. Evaluate the double integral ZZ
x2 dA
D
2 2
where D is the disk {(x, y) : x + y ≤ 9}.
   
27π 9π 81π
A 9π B C D
 4 2 4
9π
E 4

Q. 5. In R3 , let (P ) be the plane of equation 4x − 2y − 2z + 8 = 0. Which function in the

following whose graph is the plane (P )?

A f (x, y)

= 8x − 4y + 4
B f (x, y)

= 2x − y + 4
C f (x, y)

= 2x − 3y + 8
D f (x, y)

=x−y+2
E f (x, y) = 2x − 2y + 1
p
Q. 6. Let (C)
p be the intersection between the surface z = 8 − x2 − y 2 and the cone
z = x2 + y 2 . The projection of (C) onto the plane Oxy is a circle. Evaluate its
radius.
 √  
A 1 B 2 3 C 2 D 4
√
E 2 2

Q. 7. Find the maximum value of the function f (x, y) = 13 − x2 − y 2 for (x, y) in the
domain 
(x, y) : x4 + y 6 ≤ 1 .
 √   
A 14 2 B 13 C 12 D 11
 √
E 12 2

Q. 8. Evaluate the double integral ZZ

3dA
D

where D is the triangular region with vertices: A (−3, 0) , B (0, 5) and C (7, 0).
   
A 150 B 75 C 25 D 50

E 30

Q. 9. The points given in the following are represented in polar coordinates. Which point
is inside the circle x2 + y 2 = 4?





A 5, − π6 B 6, π4 C 32 , π6 D 5, π3


E 3, π2
2 −y 2
Q. 10. Which number from the following is in the range of the function f (x, y) = ex ?
   
A −2.71 B −0.4 C 0 D −0.3

E 0.2

2 CODE: 1765
Q. 11. In R2 , what is the greatest distance from a point on the circle x2 + y 2 = 1 to the
point M (−1, 7)?
p √ √ p √
A
p 51 + 10 2 B
p 51 C 50 + 53
 √  √
D 50 + 18 2 E 51 + 53

Q. 12. In R3 , let (C) be the intersection between the cylinder x2 + y 2 = 5 and the plane
2x + 3y − 4z + 8 = 0. Which point from the following is on the curve (C)?
   
A M (1, 2, −1) B M (−1, −2, 1) C M (1, 2, 1) D M (−1, 2, 1)

E M (1, −2, 1)

Q. 13. Let u (x, y) be a function such that

ux (−1, 0) = −4, uy (−1, 0) = −2, u (−1, 0) = 3.

Consider the function F

 
(x, y) := cos (u (x, y)). 
Evaluate Fx (−1, 0). 
A cos(4) B 6 sin(3) + cos(4) C 4 sin(3) D 6 sin(3)

E 4 sin(3) + cos(4)

Q. 14. Evaluate the value of the function

 3 10  
3x − xyz 4 − 2xyz 1
g (x, y, z) = ln
(x2 + y 4 + z 6 )8 + 1
2
ex−(y−2) −z3

at M (x = 0, y = 2, z = 0).
   
A −2 ln(2) B −3 ln(2) C 0 D 3 ln(2)

E 2 ln(2)

Q. 15. Let M be a point different from the origin and on the intersection between the line
y − x = 0 and the parabola x − y 2 = 0. Which one from the following is the polar
coordinates of M (in the standard polar coordinate system x = r cos φ, y = r sin φ)?
   √ 
A M (1, π3 ) B M (1, π4 ) C M ( 2, π4 ) D M (1, − π4 )
 √
E M ( 2, − π4 )

Q. 16. Let (S) be the paraboloid z = 10 − (x − 1)2 − (y − 2)2 , and let M be the highest
point (relative to the plane Oxy) on (S). Let (P ) be the tangent plane to (S) at M .
Which point from the following is on the plane (P )?
  
A N (1, 2, 9) B N (3, 4, 10) C N (−1, 2, 9)
 
D N (−1, −2, 9) E N (1, −2, −9)

Q. 17.How many saddle points

does the function f (x,y) = 2022 + 2021ye−x have?

A 0 B Infinitely many C 1 D 2

E 3
  
1
Q. 18. Choose a point in the domain of the function f (x, y) = ln sin x2 +y2 from the
following.
  
A M (−0.3, −0.3) B M (0.5, −0.2) C M (0.1, −0.3)
 
D M (0.2, −0.2) E M (0.2, −0.3)

3 CODE: 1765
Q. 19. Which function from the following has a local minimum at the point M (1, 0)?

A f (x, y)

= (x − 1)2 + y 2 + 3
B f (x, y)

= (x − 1)2 y + 3
C f (x, y)

= − (x − 1)2 − y 2 + 3
D f (x, y)

= (x − 1) y + 3
E f (x, y) = (x − 1) y 2 + 3
Q. 20. Among the following directions, which one that the function f (x, y) =
1
x2 +y 4
sin (x2 + y 4 ) goes uphill most rapidly at M (−1, 1)?
   
A →−u (1, 0) B → −u (4, −6) C →−u (−2, 5) D →
−
u (1, −2)

→
−
E u (3, −1)

4 CODE: 1765
 HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

—————————————————
ANSWER KEYS
SUBJECT: CALCULUS 2
DURATION: 50 MINUTES
 NO MATERIALS ARE ALLOWED
CODE: 1765
    
Q. 1. C Q. 5. B Q. 9. C Q. 13. C Q. 17. A
    
Q. 2. D Q. 6. C Q. 10. E Q. 14. C Q. 18. E
    
Q. 3. E Q. 7. B Q. 11. A Q. 15. C Q. 19. A
    
Q. 4. D Q. 8. B Q. 12. E Q. 16. B Q. 20. D

1 CODE: 1765
 Lecturer: Approved by:
CODE: 6
..............................................................................................................

Semester/ Academic year 213 2021 - 2022

MIDTERM EXAM
Date 10 July 2022
UNIVERSITY OF TECHNOLOGY Course title Calculus 2
VNUHCM Course ID MT1005
FACULTY OF AS Duration 50 mins Question sheet code: 1766

-This is a closed book exam. Total available score: 10 points.

-At the beginning of the working time, you MUST fill in your full name and student ID on this question
sheet.

Student’s full name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Student ID: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 1:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 2:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I. Multiple choice (6 points, 60 minutes)

There are 20 questions.
Zx Each question is assigned 0.5 point. (L.O.1)
d p
Question 01. If f ( x ) = cos(t2 )dt, find A = f ( x ).
dx
0
cos x sin x2 cos x sin x
A= p
Q. 1.A Find the maximumB valueA= p of the function C Af=(x,py) = 13 − x D −A y=2 for
p (x, y) in the
3 x 3 x 2 x 2 x
domain  that 4
Question 02. Find the approximation of c such
(x, y) : x + y 6 ≤ 1 .
Z x
 √  f (t)dt = x3 + 3x2 + 2x 3. 
A 14 2 B 12 c C 13 D 11
 √
E 12 2
A c = 0.9717 B c = 0.8717 C c = 0.6717 D c = 0.7717

x2 is the e2xthe
2 2
Q. Question
2.0 Which If F ( x ) =
03. number from an antiderivative
following is of in
the the
function
rangef ( x )of . Findfunction
the antiderivative
f (x, y)of=theexfunction
−y
?
f 2x
( x )e .  Z
 
A A−2.71 B 0 C −0.4
f 0 ( x )e2x dx = x2 + x + C
D −0.3
 None of them B
Z
E 0.2 Z
2x 2
0
C f ( x )e dx = x 2x + C D f 0 ( x )e2x dx = 2x2 + 2x + C

Q. 3. Let M be
a point on the circle (x + 2)2 +y 2 = 4.lnGiven x that M has polar coordinates
Question 04.3π If F ( x ) is an antiderivative of the function f ( x ) = · Calculate I = F (e) F (1).
r0 , − 4 (in the standard polar coordinate system x x = r cos φ, y = r sin φ). What
1 1
I=
A is the value of r0 ? B I = e C None of them D I=
√ e √  √ 2 
A 3 2 B 2 2 C 2 D Z 42 0

Question 05. If the functions f and g have continuous derivative on [0, 2] and satisfies the condition f ( x ) g( x ) dx =
E Z22 Z 2h i0 0

2, f ( x ) g0 ( x ) dx = 3, then calculate f ( x ) g( x ) dx.

0 0
Q. 4. The points given in the following are represented in polar coordinates. Which point
2 2
Stud. is inside the circle x + y = 4?
 Fullname:


Page 1/5 - Question sheet
code 1254


π 3 π π
A 5, − 6 B 2, 6 C 6, 4 D 5, π3


E 3, π2

1 CODE: 1766
 Q. 5. Evaluate the double integral ZZ
x2 dA
D
2 2
where D is the disk {(x, y) : x + y ≤ 9}.
   
9π 27π 81π
A 9π B C D
 2 4 4
9π
E 4

Q. 6. In R3 , let (P ) be the plane of equation 4x − 2y − 2z + 8 = 0. Which function in the

following whose graph is the plane (P )?

A f (x, y)

= 8x − 4y + 4
B f (x, y)

= 2x − 3y + 8
C f (x, y)

= 2x − y + 4
D f (x, y)

=x−y+2
E f (x, y) = 2x − 2y + 1
Q. 7. Which function from the following has a local minimum at the point M (1, 0)?

A f (x, y)

= (x − 1)2 + y 2 + 3
B f (x, y)

= − (x − 1)2 − y 2 + 3
C f (x, y)

= (x − 1)2 y + 3
D f (x, y)

= (x − 1) y + 3
E f (x, y) = (x − 1) y 2 + 3

Q. 8. In R2 , what is the greatest distance from a point on the circle x2 + y 2 = 1 to the

point M (−1, 7)?
p √ p √ √
A
p 51 + 10 2 B
p 50 + 53 C 51
 √  √
D 50 + 18 2 E 51 + 53
Q. 9. Evaluate the value of the function
 3 10  
3x − xyz 4 − 2xyz 1
g (x, y, z) = ln
(x2 + y 4 + z 6 )8 + 1
2
ex−(y−2) −z3

at M (x = 0, y = 2, z = 0).
   
A −2 ln(2) B 0 C −3 ln(2) D 3 ln(2)

E 2 ln(2)
p
Q. 10. Let (C)
p be the intersection between the surface z = 8 − x2 − y 2 and the cone
z = x2 + y 2 . The projection of (C) onto the plane Oxy is a circle. Evaluate its
radius.
  √ 
A 1 B 2 C 2 3 D 4
√
E 2 2

2 CODE: 1766
Q. 11. Let D = {(x, y) : x2 + y 2 ≤ 1, y ≤ −x}. In the polar coordinate system x =
r cos (φ) , y = r sin (φ), what is the range for (r, φ) corresponding to the domain
D?
Choose the correct statement from the following.

A 0 ≤ r ≤ 1, −π ≤ φ ≤ π

B 0 ≤ r ≤ 1, − π4 ≤ φ ≤ 3π
 4
C 0 ≤ r ≤ 1, 3π ≤φ≤ 3π
 4 2
D 0 ≤ r ≤ 1, − 5π ≤ φ ≤ − π4
 4
E 0 ≤ r ≤ 1, − π4 ≤ φ ≤ π
4
  
1
Q. 12. Choose a point in the domain of the function f (x, y) = ln sin x2 +y 2 from the
following.
  
A M (−0.3, −0.3) B M (0.1, −0.3) C M (0.5, −0.2)
 
D M (0.2, −0.2) E M (0.2, −0.3)

Q. 13. Let (S) be the paraboloid z = 10 − (x − 1)2 − (y − 2)2 , and let M be the highest
point (relative to the plane Oxy) on (S). Let (P ) be the tangent plane to (S) at M .
Which point from the following is on the plane (P )?
  
A N (1, 2, 9) B N (−1, 2, 9) C N (3, 4, 10)
 
D N (−1, −2, 9) E N (1, −2, −9)

Q. 14. Among the following directions, which one that the function f (x, y) =
1
x2 +y 4
sin (x2 + y 4 ) goes uphill most rapidly at M (−1, 1)?
   
A →−u (1, 0) B → −u (−2, 5) C →−u (4, −6) D →
−
u (1, −2)

→
−
E u (3, −1)

Q. 15.How many saddle points


does the function f (x,y) = 2022 + 2021ye−x have?

A 0 B 1 C Infinitely many D 2

E 3

Q. 16. In R3 , let (C) be the intersection between the cylinder x2 + y 2 = 5 and the plane
2x + 3y − 4z + 8 = 0. Which point from the following is on the curve (C)?
   
A M (1, 2, −1) B M (1, 2, 1) C M (−1, −2, 1) D M (−1, 2, 1)

E M (1, −2, 1)

Q. 17. Let g be the function defined by

 10  
3x3 − xyz 4 − 2xyz 1
g (x, y, z) = ln .
(x2 + y 4 + z 6 )8
2
ex−(y−2) −z 3

Evaluate gy (−1, 0, 0).

   
A −4245690 B −3753892 C −2344900 D 1845654

E −4194304

3 CODE: 1766
Q. 18. Let u (x, y) be a function such that

ux (−1, 0) = −4, uy (−1, 0) = −2, u (−1, 0) = 3.

Consider the function F

 
(x, y) := cos (u (x, y)). 
Evaluate Fx (−1, 0). 
A cos(4) B 4 sin(3) C 6 sin(3) + cos(4) D 6 sin(3)

E 4 sin(3) + cos(4)

Q. 19. Evaluate the double integral ZZ

3dA
D

where D is the triangular region with vertices: A (−3, 0) , B (0, 5) and C (7, 0).
   
A 150 B 25 C 75 D 50

E 30

Q. 20. Let M be a point different from the origin and on the intersection between the line
y − x = 0 and the parabola x − y 2 = 0. Which one from the following is the polar
coordinates of M (in the standard polar coordinate system x = r cos φ, y = r sin φ)?
  √  
A M (1, π3 ) B M ( 2, π4 ) C M (1, π4 ) D M (1, − π4 )
 √
E M ( 2, − π4 )

4 CODE: 1766
 HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

—————————————————
ANSWER KEYS
SUBJECT: CALCULUS 2
DURATION: 50 MINUTES
 NO MATERIALS ARE ALLOWED
CODE: 1766
    
Q. 1. C Q. 5. D Q. 9. B Q. 13. C Q. 17. E
    
Q. 2. E Q. 6. C Q. 10. B Q. 14. D Q. 18. B
    
Q. 3. B Q. 7. A Q. 11. D Q. 15. A Q. 19. C
    
Q. 4. B Q. 8. A Q. 12. E Q. 16. E Q. 20. B

1 CODE: 1766
 Lecturer: Approved by:
CODE: 7
..............................................................................................................

Semester/ Academic year 213 2021 - 2022

MIDTERM EXAM
Date 10 July 2022
UNIVERSITY OF TECHNOLOGY Course title Calculus 2
VNUHCM Course ID MT1005
FACULTY OF AS Duration 50 mins Question sheet code: 1767

-This is a closed book exam. Total available score: 10 points.

-At the beginning of the working time, you MUST fill in your full name and student ID on this question
sheet.

Student’s full name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Student ID: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 1:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Invigilator 2:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I. Multiple choice (6 points, 60 minutes)

There are 20 questions.
Zx Each question is assigned 0.5 point. (L.O.1)
d p
Question 01. If f ( x ) = cos(t2 )dt, find A = f ( x ).
dx
0
cos x sin x cos x sin x
A = points
Q. 1.A The p given inBthe A =following
p C A = p in polar coordinates.
are represented D A = p Which point
3 x 2 23 x 2 x 2 x
is inside the circle x + y = 4?

Question  of

02.
Find the approximation c such that 


A 5, − π6 B 32 , π6 Z x C 5, π3 D 6, π4


E 3, π2 f (t)dt = x3 + 3x2 + 2x 3.
c

Q. 2.A Find
c = 0.9717 the maximumB value of the function
c = 0.8717 C c f=(x,
0.6717 y) = 13 − x2D −c =y 20.7717
for (x, y) in the
domain  of the function
Question 03. If F ( x ) = x2 is an antiderivative 4
2x
6 f ( x )e . Find the antiderivative of the function
0 2x
f ( x )e . (x, y) : x + y ≤ 1 .
Z
 √ of them
A None  B f 0 ( x)e2x dx = x2 + x + C 
A 14Z 2 B 12 Z C 11 D 13
C √ f 0 ( x )e2x dx = x2 2x + C D f 0 ( x )e2x dx = 2x2 + 2x + C
E 12 2
ln x
Q. Question
3. In R04.
3 If F(C)
, let ( x ) isbe
an antiderivative of the function
the intersection between f ( x )the · Calculate
= cylinder +Fy(e2) = F5(1)and
x2I = . the plane
x
1 1
I =+ 3y − 4z + 8 =B0.I Which
A 2x =e point fromC the None following
of them is on the D I= curve (C)?
 e   2
A M (1, 2, −1) B M (1, 2, 1) C M (−1, 2, 1) Z 2
 
Question 05. If the functions f and g have continuous derivative on [0, 2] and satisfies the condition f 0 ( x ) g( x ) dx =
D ZM (−1, −2, 1) E MZ(1,h −2, 1) i 0
2 2 0
2, f ( x ) g0 ( x ) dx = 3, then calculate f ( x ) g( x ) dx.
3
Q. 4. 0In R , let (P ) be the plane0 of equation 4x − 2y − 2z + 8 = 0. Which function in the
following whose graph is the plane (P )?
Stud. Fullname: Page 1/5 - Question sheet code 1254

A f (x, y)

= 8x − 4y + 4
B f (x, y)

= 2x − 3y + 8
C f (x, y)

=x−y+2
D f (x, y)

= 2x − y + 4
E f (x, y) = 2x − 2y + 1

1 CODE: 1767
 Q. 5. Evaluate the double integral ZZ
x2 dA
D
2 2
where D is the disk {(x, y) : x + y ≤ 9}.
   
9π 81π 27π
A 9π B C D
 2 4 4
9π
E 4

Q. 6. Let (S) be the paraboloid z = 10 − (x − 1)2 − (y − 2)2 , and let M be the highest
point (relative to the plane Oxy) on (S). Let (P ) be the tangent plane to (S) at M .
Which point from the following is on the plane (P )?
   
A N (1, 2, 9) B N (−1, 2, 9) C N (−1, −2, 9) D N (3, 4, 10)

E N (1, −2, −9)

Q. 7. Let D = {(x, y) : x2 + y 2 ≤ 1, y ≤ −x}. In the polar coordinate system x =

r cos (φ) , y = r sin (φ), what is the range for (r, φ) corresponding to the domain
D?
Choose the correct statement from the following.

A 0 ≤ r ≤ 1, −π ≤ φ ≤ π

B 0 ≤ r ≤ 1, − π4 ≤ φ ≤ 3π
 4
C 0 ≤ r ≤ 1, − 5π ≤ φ ≤ − π4
 4
D 0 ≤ r ≤ 1, 3π ≤φ≤ 3π
 4 2
E 0 ≤ r ≤ 1, − π4 ≤ φ ≤ π4

Q. 8. Let u (x, y) be a function such that

ux (−1, 0) = −4, uy (−1, 0) = −2, u (−1, 0) = 3.

Consider the function F

 
(x, y) := cos (u (x, y)). 
Evaluate Fx (−1, 0).
A cos(4) B 4 sin(3) C 6 sin(3)
 
D 6 sin(3) + cos(4) E 4 sin(3) + cos(4)

Q. 9. Let M be
a point on the circle (x + 2)2 +y 2 = 4. Given that M has polar coordinates
r0 , − 3π
4
(in the standard polar coordinate system x = r cos φ, y = r sin φ). What
is the value of r0 ?
√ √  √
A 3 2 B 2 2 C 4 D 2

E 2

Q. 10. Evaluate the double integral ZZ

3dA
D

where D is the triangular region with vertices: A (−3, 0) , B (0, 5) and C (7, 0).
   
A 150 B 25 C 50 D 75

E 30

2 CODE: 1767
Q. 11. In R2 , what is the greatest distance from a point on the circle x2 + y 2 = 1 to the
point M (−1, 7)?
p √ p √ p √ √
A
p 51 + 10 2 B 50 + 53 C 50 + 18 2 D 51
 √
E 51 + 53
  
1
Q. 12. Choose a point in the domain of the function f (x, y) = ln sin x2 +y2 from the
following.
  
A M (−0.3, −0.3) B M (0.1, −0.3) C M (0.2, −0.2)
 
D M (0.5, −0.2) E M (0.2, −0.3)

Q. 13.How many saddle points


does the function f (x,y) = 2022 + 2021ye−x have?
A 0 B 1 C 2
 
D Infinitely many E 3

Q. 14. Evaluate the value of the function

 3 10  
3x − xyz 4 − 2xyz 1
g (x, y, z) = ln
(x2 + y 4 + z 6 )8 + 1
2
ex−(y−2) −z3

at M (x = 0, y = 2, z = 0).
   
A −2 ln(2) B 0 C 3 ln(2) D −3 ln(2)

E 2 ln(2)

Q. 15. Which function from the following has a local minimum at the point M (1, 0)?

A f (x, y)

= (x − 1)2 + y 2 + 3
B f (x, y)

= − (x − 1)2 − y 2 + 3
C f (x, y)

= (x − 1) y + 3
D f (x, y)

= (x − 1)2 y + 3
E f (x, y) = (x − 1) y 2 + 3
Q. 16. Let g be the function defined by
 10  
3x3 − xyz 4 − 2xyz 1
g (x, y, z) = ln .
(x2 + y 4 + z 6 )8
2
ex−(y−2) −z 3

Evaluate gy (−1, 0, 0).

   
A −4245690 B −3753892 C 1845654 D −2344900

E −4194304
2 −y 2
Q. 17. Which number from the following is in the range of the function f (x, y) = ex ?
   
A −2.71 B 0 C −0.3 D −0.4

E 0.2

3 CODE: 1767
 p
Q. 18. Let (C)
p be the intersection between the surface z = 8 − x2 − y 2 and the cone
2 2
z = x + y . The projection of (C) onto the plane Oxy is a circle. Evaluate its
radius.
   √
A 1 B 2 C 4 D 2 3
√
E 2 2

Q. 19. Among the following directions, which one that the function f (x, y) =
1
x2 +y 4
sin (x2 + y 4 ) goes uphill most rapidly at M (−1, 1)?
   
A →−u (1, 0) B → −u (−2, 5) C →−u (1, −2) D →
−
u (4, −6)

→
−
E u (3, −1)

Q. 20. Let M be a point different from the origin and on the intersection between the line
y − x = 0 and the parabola x − y 2 = 0. Which one from the following is the polar
coordinates of M (in the standard polar coordinate system x = r cos φ, y = r sin φ)?
  √  
A M (1, π3 ) B M ( 2, π4 ) C M (1, − π4 ) D M (1, π4 )
 √
E M ( 2, − π4 )

4 CODE: 1767
 HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

—————————————————
ANSWER KEYS
SUBJECT: CALCULUS 2
DURATION: 50 MINUTES
 NO MATERIALS ARE ALLOWED
CODE: 1767
    
Q. 1. B Q. 5. C Q. 9. B Q. 13. A Q. 17. E
    
Q. 2. D Q. 6. D Q. 10. D Q. 14. B Q. 18. B
    
Q. 3. E Q. 7. C Q. 11. A Q. 15. A Q. 19. C
    
Q. 4. D Q. 8. B Q. 12. E Q. 16. E Q. 20. B

1 CODE: 1767