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Lecturer: Huỳnh Thị Hồng Diễm . . . Approved by: Lê Xuân Đại . . . . . . . . .
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UNIVERSITY OF
TECHNOLOGY
Faculty of AS
Final Exam Semester/Academic year I 2025-2026
Date 23/12/2025
Course title: Calculus 1
Course ID: MT1003
Duration: 70 Mins Code: 2303
Notes: - You are NOT allowed to use materials.
- Each correct answer is awarded 10/22 points.
- Each incorrect answer is deducted by 25% of correct answer.
- Answers left blank will not be counted.
- Do not round between steps. Round your final answers to 4 decimal places.
Question 1. (L.O.1) Calculate the average value of f(x) = x
x
2 + 1
, where −1 ≤ x ≤ 2.
A. −0.0333 B. −0.05 C. 0.4581 D. 0.1527
Question 2. (L.O.2) Let f(x) = 10
1 − e
−
t
3
. Find f
−1
(9).
A. −3 ln
9
10
B. − ln
11
10
C. 3 ln
9
10
D. 3 ln(10)
Question 3. (L.O.1) Let f(x) be a function satisfying f(6.3) = 6.7, f
′
(6.3) = 5. Using the
linear approximation to estimate the value of f(5.985), we obtain
A. 5.1250 B. 2.8895 C. 8.275 D. 7.1105
Question 4. The shape of a fountain pond is shown in the figure below. Assuming the
thickness of the walls and bottom of the pond is neglected, the volume of the pond is equal
to the volume of the solid of revolution generated by revolving the region bounded by the
curve y =
−(x + 2)3
2
+
1
2
, 2 coordinate axes and the line y = 1 about y-axis. Given that the
units on the coordinate axes are in meters (m), the volume of the pond is equal to
y
x
1
y
x
A. 14.4513 m3 B. 13.8230 m3 C. 7.6296 m3 D. 16.9646 m3
Question 5. (L.O.1) Let f(x) = Z 5x+4
3
t
3
p
3 + t
2 d t be a function. Find f
′
(−1).
A. −2 B. 2 C. −10 D. 0
Question 6. (L.O.1) Find the general solution of the differential equation
y
′ = x
2
y
2 + 9x
2 + 4y
2 + 36.
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A. y = 3 tan
x
3 + 12x
+ C B. y = 3 tan
x
3 + 12x + C
C. y = tan
3 tan
x
3 + 12x
+ C D. y = tan
x
3 + 12x + C
Question 7. (L.O.2) Let y = f(x) be a function whose graph is shown in the figure below.
y
x
y = f(x)
−10 2
−2
2
3
2
1
Let g(x) = Z x
−10
f(t) d t. Given that the areas of regions 1, 2 and 3 are 1.3, 3.5 and 9.6,
respectively. Find the maximum value of g(x), where −10 ≤ x ≤ 2.
A. −7.4 B. None of the others
C. −6.1 D. 1.5
Question 8. (L.O.1) Given the parametric curve:
(
x = t
2 + 1,
y =
√
t + 2.
The equation of the tangent line to this curve at (2, 1) is
A. 4
√
3x − y = 4√
3 − 2 B. 4x + y = 6
C. The tangent line does not exist D. 4
√
3x − y = 10
Question 9. (L.O.2) Suppose the curve in the figure below is the graph of a particular
solution of one of the four differential equations below. Choose the correct equation.
y
x
A. y
′ =
−x
2
y
B. y
′ = −3y C. y
′ = 2y D. y
′ = −x
2
y
Question 10. Given that the length of the curve y = cosh x on the interval [0, ln 2] is a
4
.
Determine a.
A. 5 B. 3
2
C. 6 D. 3
Question 11. (L.O.1) Let y(x) be the solution of the differential equation: y
′ +
2y
x
=
3, y(1) = 1.8, then:
A. y(2) = 2.5 B. y(2) = 2.575 C. y(2) = 2.8 D. y(2) = 2.2
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Question 12. (L.O.2) Let F(x) be a function satisfying F
′
(x) = arcsin2 x
√
1 − x
2
. Given that
lim
x→1−
F(x) = 0. Find F(−0.9)
A. −1.2319 B. −1.7599 C. −1.0559 D. −2.1119
Question 13. (L.O.1) The total number of asymptotes (vertical, horizontal, slant) of the
curve y =
x − 2
√
4 − x
2
is
A. 1 B. 0 C. 3 D. 2
Question 14. (L.O.2) Let F(x) be a function satisfying F
′
(x) = 1
x
2 + x + 1
. Given that
lim
x→+∞
F(x) = 0. Find F(2).
A. −0.4621 B. −0.3851 C. −0.4236 D. −0.2311
Question 15. (L.O.1) Use the central Riemann sum with a uniform 10-segment partition
of [0, 1] to approximate Z 1
0
x arctan x d x.
A. 0.2472 B. 0.2850 C. 0.2854 D. 0.2849
Question 16. (L.O.2) Given the function y = y(x) defined by the parametric equation
x = sin t − 2t, y = 2 (1 − cos(2t)).
On which of the following intervals is this function decreasing?
A.
π
2
,
3π
4
B.
−
π
4
,
π
4
C.
3π
4
,
5π
4
D.
0,
π
4
Question 17. (L.O.2) Let y =
4
x
2
, where x < 0. Consider rectangles with vertices
(0, 0),(x, 0),(0, y),(x, y), where (x, y) belongs to the graph of the above function. Find the
vertex (x, y) of the rectangle for which the perimeter is minimized.
A. x = −2, y = 1 B. x = −3, y = 1 C. x = −1, y = 4 D. x = −4, y =
1
4
Question 18. Find all values of a such that
Z∞
0
(x + 1)e
axdx = 2
A. a = 1/2 B. a = −1
C. a = −1 or a = 1/2 D. a does not exist
Question 19. (L.O.2) Let F(x) be an antiderivative of the function ln x
√3 x
, satisfying F(1) =
−2.25. Which of the following statements is true?
A. lim
x→0+
F(x) = 1 B. lim
x→0+
F(x) = 0
C. All other statements are wrong. D. lim
x→0+
F(x) = +∞
Using the following information, answer the questions from 20 to 22.
The current intensity I(t) (A - ampere) in an LC circuit consisting of an
inductor L (H - henry), a capacitor C (F - farad) and connected to a power
source with voltage E(t) (V - volt) satisfies the differential equation
L · I
′′(t) + 1
C
· I(t) = E
′
(t).
Consider the LC circuit (the switch is temporarily not closed), where
L = 10H, C = 0.00625F, connected to a voltage source E(t).
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Question 20. (L.O.2) Suppose that E(t) is a constant and when the switch is closed,
the current intensity in the circuit satisfies I(0) = 2, I′
(0) = 1, calculate the magnitude of
I(3)
A. 0.5543 A B. 2.5541 A C. 0.2455 A D. 1.5536 A
Question 21. (L.O.2) When connecting the above circuit to a voltage source E(t) =
110 sin (20t) V and closing the switch, a current will appear in the circuit that does not
gradually die out over time (called a steady oscillating current), denoted Isp. This current
is the particular solution (can be found by the method of undetermined coefficients) of
the above differential equation. With some constants A and B, the form of the particular
solution is
A. A sin(20t) B. t(A cos(20t) + B sin(20t))
C. A cos(20t) + B sin(20t) D. Atsin(20t)
Question 22. (L.O.2) Calculate the magnitude of Isp(1.5).
A. 0.8230 A B. 0.0832 A C. 0.0884 A D. 0.3228 A
END
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