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Calculus

Instelling
Calculus

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.................................................................................................................

Semester/ Academic year 221 2022-2023
FINAL EXAM
Date 26 December 2022
UNIVERSITY OF TECHNOLOGY Course title Calculus 1
VNUHCM Course ID MT1003
FACULTY OF AS Duration 1234 100 mins Question sheet code
Intructions to students: - There are 14 pages in the exam
-This is a closed book exam. Only your calculator is allowed. Total available score: 10.
-For multiple choice questions, you get 0.5 for a correct answer, loose 0.1 for a wrong answer,
no deduction unanswered questions. You choose a correct answer with a tolerance of 0.005 for each question.
-At the beginning of the working time, you MUST fill in your full name and student ID on this question sheet.
-All essential steps of calculations, analyses, justifications and final results are required for full credit.
Any answer without essential calculation steps, and/or analyses, and/or justifications will earn zero mark.

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

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

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

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

Part I. Multiple choice (6 points, 60 minutes)
Z x
Question 01. [L.O.1.1] Identify all local extrema of f ( x ) = (t2 − 3t + 2)dt.
0
A None of them B (1, 5/6) and (2, 2/3) C (1, 7/6) and (2, 2/3)
D (1, 5/6) E (1, 5/6) and (2, 3/2)
Z x
Question 02. [L.O.1.2] Find the maximum and minimum values of f ( x ) = (3t2 − 6t − 9)dt on the interval
−2
[−2, 2].
A f max = 6; f min = −21 B None of them C f max = 7; f min = −20
D f max = 7; f min = −21 E f max = 6; f min = −20

Zx
πt2

Question 03. [L.O.1.2] If f ( x ) = sin dt, find all absolute extrema of f on the interval [0, 3].
2
0 √ √
A f max = f (0); f min = f (2) B f max = f ( 2); f min = f ( 8) C None of them
√ √ √
D f max = f ( 2); f min = f ( 6) E f max = f ( 2); f min = f (2)

Zx
Question 04. [L.O.1.2] If f ( x ) = (4t3 − 4t)dt, find all absolute extrema of f on the interval [−1, 2].
−1
A f max = 9; f min = 1 B f max = 9; f min = 0 C f max = 8; f min = 1
D None of them E f max = 8; f min = 0
3
Zx
dt
Question 05. [L.O.1.1] Find the derivative of the function f ( x ) = √ ·
2 + t3
arctan x

Stud. Fullname: Page 1/14 - Question sheet code 1234

, 2x2 1 1 3x2 1 1
A √ −p · B √ −p ·
2+x 9 2 + (arctan x ) 1 + x2
3 2+ 4+x9 1 + x2
(arctan x )3
3x 2 1 1 3x 2 1 1
C √ −p · D √ −p ·
2 + x9 3 + (arctan x )3 1 + x2 2 + x9 2 + (arctan x )3 1 + x2
E None of them

Z x
sin
p Zy π
Question 06. [L.O.1.1] If f ( x ) = 1 + t2 dt and g(y) = f ( x )dx, find A = g′′
6
√ √0 √ 0
√
11 17 13 15
A A= B A= C A= D None of them E A=
4 4 4 4
Zx
d √
Question 07. [L.O.1.1] If f ( x ) = cos(t2 )dt, find A = f ( x ).
dx
0
sin x cos x cos x sin x
A A= √ B A= √ C None of them D A= √ E A= √
3 x 2 x 3 x 2 x
Z x
Question 08. [L.O.1.1] Find all values of c such that f (t)dt = x2 + x − 2.
c
A c = −2 B None of them C c = 1 or c = −2 D c = −2 or c = 0 E c=1
Z x
Question 09. [L.O.1.1] Find the approximation of c such that f (t)dt = x3 + 3x2 + 2x − 3.
c
A None of them B c = 0.7717 C c = 0.8717 D c = 0.6717 E c = 0.9717

t2
Z x
Question 10. [L.O.1.2] On what interval is the curve y = dt concave downward?
0 t2 + t + 2
A (−2, 0) B None of them C (−4, 0) D [−2, 0] E [−4, 0]

Question 11. π [L.O.2.1] Find the antiderivative F ( x ) of the function f ( x ) = sin x + cos x which satisfies the
condition F = 2.
2
A F ( x ) = − cos x + sin x + 3 B F ( x ) = − cos x + sin x + 1 C F ( x ) = − cos x + sin x − 3
D F ( x ) = − cos x + sin x − 1 E None of them

Question 12. [L.O.2.1] If F ( x ) is an antiderivative of the function f ( x ) = ex + 2x which satisfies the condition
3
F (0) = · Find F ( x ).
2
1 1
A F ( x ) = ex + x2 + B F ( x ) = 2ex + x2 − C None of them
2 2
1 5
D F ( x ) = ex + x2 − E F ( x ) = ex + x2 +
2 2
1 f (x)
Question 13. [L.O.2.1] If F ( x ) = is an antiderivative of the function · Find the antiderivative of the
2x2 x
′
function f ( x ) ln x.

ln x 1 ln x 1
Z Z
′ ′
A f ( x ) ln x dx = 2 + 2 + C B f ( x ) ln x dx = − 2
+ 2 +C
x x x 2x
ln x 1 ln x 1
Z Z
C f ′ ( x ) ln x dx = − 2
+ 2 +C D f ′ ( x ) ln x dx = − 2
− 2 +C
x x x x
E None of them

1 f (x)
Question 14. [L.O.2.1] If F ( x ) = − 3
is an antiderivative of the function · Find the antiderivative of the
3x x
function f ′ ( x ) ln x.

Stud. Fullname: Page 2/14 - Question sheet code 1234

, ln x 1
Z
A None of them B f ′ ( x ) ln x dx =
3
− 5 +C
x 5x
ln x 1 ln x 1
Z Z
C f ′ ( x ) ln x dx = − 3 + 3 + C D ′
f ( x ) ln x dx = − 3 − 3 + C
x 3x x 3x
ln x 1
Z
E f ′ ( x ) ln x dx = 3 + 3 + C
x 3x
Question 15. [L.O.2.1] If F ( x ) = x2 is an antiderivative of the function f ( x )e2x . Find the antiderivative of the
function
Z
f ′ ( x )e2x . Z
′
A 2x 2
f ( x )e dx = −2x + 2x + C B f ′ ( x )e2x dx = x2 − 2x + C
Z
C f ′ ( x )e2x dx = −2x2 − 2x + C D None of them
Z
E f ′ ( x )e2x dx = − x2 + x + C

ln x
Question 16. [L.O.2.1] If F ( x ) is an antiderivative of the function f ( x ) = · Calculate I = F (e) − F (1).
x
1 1
A I=2 B I= C None of them D I= E I=e
2 e
Question 17. [L.O.2.1] Find the function f given that the slope of the tangent line to the graph of f at any point
ln x
( x, f ( x )) is f ′ ( x ) = √ and that the graph of f passes through the point (1, 0).
x
√ √ √ √ √ √
A f ( x ) = 3 x. ln x + 4 x + 4 B f ( x ) = 2 x. ln x − 3 x + 4 C f ( x ) = 2 x. ln x − 4 x + 4
√ √
D None of them E f ( x ) = 3 x. ln x − 4 x + 4

Question 18. [L.O.2.1] Find the function f given that the slope of the tangent line to the graph of f at any point
( x, f ( x )) is f ′ ( x ) = xe−3x and that the graph of f passes through the point (0, 0).
xe−3x e−3x 1 xe−3x e−3x 1
A f ( x ) = −2 − + B f ( x ) = −2 − + C None of them
3 9 9 3 3 9
xe − 3x e − 3x 1 xe − 3x e − 3x 1
D f (x) = − − + E f (x) = − − +
3 3 9 3 9 9
Z π Z π
2 2
Question 19. [L.O.1.1] If f ( x ) dx = 5, then calculate I = [ f ( x ) + 2 sin x ] dx.
0 0
π
A 3 B 7 C None of them D 5+ E 5+π
2
Z 2 Z 2 Z 2 h i
Question 20. [L.O.1.1] If f ( x ) dx = 2 and g( x ) dx = −1, then calculate I = x + 2 f ( x ) − 3g( x ) dx.
−1 −1 −1
11 7 17 5
A I= B I= C I= D None of them E I=
2 2 2 2
Z 6 Z 2
Question 21. [L.O.1.1] If f ( x )dx = 12, then calculate I = f (3x )dx
0 0
A I=4 B I=6 C I = 36 D I=2 E None of them

Z1
1 1
Question 22. [L.O.1.1] If − dx = a ln 2 + b ln 3 where a, b are integers. Which statement is al-
x+1 x+2
0
ways true?
A a+b = 2 B a + 2b = 0 C a + b = −2 D a − 2b = 0 E None of them

Question 23. [L.O.1.1] If the function f ( x ) has continuous derivative on [0, 1] and satisfies the condition 2 f ( x ) +
√ Z 1
3 f (1 − x ) = 1 − x2 then calculate f ′ ( x ) dx.
0

Stud. Fullname: Page 3/14 - Question sheet code 1234

, 3 1
A I=0 B I=1 C I= D None of them E I=
2 2
Question 24. [L.O.1.1] If the function f ( x ) has continuous derivative on [0, 1] and satisfies the condition f (0) =
Z 1h i
x ′
f (1) = 1, and e f ( x ) + f ( x ) dx = ae + b, where a, b are integers. Calculate Q = a2018 + b2018 .
0
A Q=0 B Q=2 C None of them D Q = 22017 + 1 E Q = 22017 − 1

Question 25. [L.O.1.1] If the functions f and g have continuous derivative on [0, 2] and satisfies the condition
Z 2 Z 2 Z 2h i′
f ′ ( x ) g( x ) dx = 2, f ( x ) g′ ( x ) dx = 3, then calculate f ( x ) g( x ) dx.
0 0 0
A I=6 B None of them C I=0 D I=5 E I=1

Question 26. [L.O.1.1] Suppose f ′′ is continuous on [1, 3] and f (1) = 2, f (3) = −1, f ′ (1) = 2, and f ′ (3) = −1.
Z3
Evaluate I = x f ′′ ( x )dx.
1
A I = −1 B I=1 C I = −2 D None of them E I=0
Z x
f (t)
Question 27. [L.O.1.1] If the functions f is continuous on [ a, +∞) ( a > 0) and satisfies the condition dt +
√ a t2
6 = 2 x, then calculate f (4).
A f (4) = 8 B None of them C f (4) = 2 D f (4) = 16 E f (4) = 4
Z x2
Question 28. [L.O.1.1] If the functions f is continuous on [0, +∞) and satisfies the condition f (t) dt =
0
1
x. sin(πx ), then calculate f .
4
1 π 1 π 1 1 1
A f = 1+ B f =− C None of them D f = E f =1
4 2 4 2 4 2 4
Question 29. [L.O.2] When a particle is located a distance x meter from the origin, a force of given F ( x ) =
1 √
2
(newton) acts on it. How much work is done in moving it from x = 1 to x = 2 3 − 1.
x + 2x + 5
π 4π
A W= B W=2 C None of them D W= E W = 2π
24 3
Question 30. [L.O.2] When a particle is located a distance x meter from the origin, a force of given F ( x ) =
1 13
√ (newton) acts on it. How much work is done in moving it from x = to x = 9.
9 + 8x − x 2 2
π 2π π 4π
A W= B W= C W= D None of them E W=
4 3 3 3
Question 31. [L.O.2] When a particle is located a distance x meter from the origin, a force of given F ( x ) =
5x2 + 20x + 6
(newton) acts on it. How much work is done in moving it from x = 3 to x = 5.
x ( x + 1)2
2 × 56 2 × 56 2 × 56

3 3 5
A W = + 2 ln B W = + ln C W = + ln
4 37 4 37 4 37
6

1 2×5
D W = + ln E None of them
4 37
Question 32. [L.O.2] When a particle is located a distance x meter from the origin, a force of given F ( x ) =
2x2 − 5x + 2
(newton) acts on it. How much work is done in moving it from x = 1 to x = 5.
x3 + x
5π 5π 9π
A W = 3 ln 5 − 5 arctan 5 + B W = 2 ln 5 − 5 arctan 5 + C W = 3 ln 5 − 5 arctan 5 +
4 4 4
7π
D W = 2 ln 5 − 5 arctan 5 + E None of them
4

Stud. Fullname: Page 4/14 - Question sheet code 1234

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