Lecturer: Approved by:

CODE: 5
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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