Final August 23 2022 - Calculus 1 | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố HCM

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Môn: Calculus 1 (MA001IU)

Trường: Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

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Profile Mai Nguyệt
Calculus 1 International University, Vietnam National University-HCM Page 1 of 1
CALCULUS 1 (MA001IU) – FINAL EXAMINATION
Semester 3, 2021-22 • Duration: 120 minutes • Date: August 23, 2022
SUBJECT: CALCULUS 1
Department of Mathematics Lecturer
Nguyen Minh Quan Nguyen Anh Tu, Nguyen Minh Quan
INSTRUCTIONS:
• Use of calculator is allowed. Each student is allowed two double-sided sheets of notes (size A4
or similar). All other documents and electronic devices are forbidden.
• Write the steps you use to arrive at the answers to each question. No marks will be given for
the answer alone.
• There are a total of 10 (ten) questions. Each one carries 10 points.
1. Air is being pumped into balloon
A with the rate of 2 cm
3
/s. Concurrently, balloon B is being
inflated in such a way that its radius is always 2 cm bigger than that of balloon A. What is the rate
of change of the volume of balloon B at the moment its radius is 4 cm?
2. Find the absolute maximum and minimum values of
f (x) = (x
2
+ 2 1x − )e
−2x
on [−4 .,2]
3. Find the following limit if it exists, or show that the limit does not exist
lim
x
→0
+
x
√
x
.
4. Using Newton’s method starting with
x
1
= 0, find the root of x
4
−x
2
= 1 correct to six decimal
places.
5. Let f be a differentiable function with f (0) = 1, 2,f (1) = f (2) = 3. By considering g(x) =
f f
(x + 1) − (x), show that there exits c ∈ (0,1 1 .) so that f
′
(c + ) = f
′
(c)
6. Find the derivative of the function
H
(x) =
Z
2x+1
1
t
t
4
+ 1
dt.
7. Evaluate
Z
1
0
( )
2x + 1 e
−x
dx.
8. Determine whether the improper integral
Z
∞
2
2
x
3
+ x
x x
4
+
2
−2
dx is convergent or divergent. Explain.
9. The table below presents the dependence of the temperature of a liquid on timeT t (in minutes).
Use the Trapezoidal Rule to approximate the average temperature of this liquid during 0 5.≤t ≤
Time t (in minutes) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Temperature T (in
o
C) 95 94.3 93.5 92.8 92.1 91.3 90.6 89.9 89.2 88.5 87.9
10. Find the arc length of the curve
y = 2(x −1)
3
2
between x = 1 and 3.x =
—END OF THE QUESTION PAPER. GOOD LUCK!—
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Calculus 1
International University, Vietnam National University-HCM Page 1 of 1
CALCULUS 1 (MA001IU) – FINAL EXAMINATION
Semester 3, 2021-22 • Duration: 120 minutes • Date: August 23, 2022 SUBJECT: CALCULUS 1 Department of Mathematics Lecturer Nguyen Minh Quan
Nguyen Anh Tu, Nguyen Minh Quan INSTRUCTIONS:
• Use of calculator is allowed. Each student is allowed two double-sided sheets of notes (size A4
or similar). All other documents and electronic devices are forbidden.
• Write the steps you use to arrive at the answers to each question. No marks will be given for the answer alone.
• There are a total of 10 (ten) questions. Each one carries 10 points.
1. Air is being pumped into balloon A with the rate of 2 cm3/s. Concurrently, balloon B is being
inflated in such a way that its radius is always 2 cm bigger than that of balloon A. What is the rate
of change of the volume of balloon B at the moment its radius is 4 cm?
2. Find the absolute maximum and minimum values of f (x) = (x2 + 2x − 1)e−2x on [−4, 2].
3. Find the following limit if it exists, or show that the limit does not exist √ lim x x. x→0+
4. Using Newton’s method starting with x1 = 0, find the root of x4 − x2 = 1 correct to six decimal places.
5. Let f be a differentiable function with f (0) = 1, f (1) = 2, f (2) = 3. By considering g(x) =
f (x + 1) − f (x), show that there exits c ∈ (0, 1) so that f ′(c + 1) = f ′(c).
6. Find the derivative of the function Z 2x+1 t H(x) = dt. 1 t4 + 1 Z 1 7. Evaluate (2x + 1)e−x dx. 0 Z ∞ 2x3 +x
8. Determine whether the improper integral
dx is convergent or divergent. Explain. 4 2 2 x + x − 2
9. The table below presents the dependence of the temperature T of a liquid on time t (in minutes).
Use the Trapezoidal Rule to approximate the average temperature of this liquid during 0 ≤ t ≤ 5. Time t (in minutes) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Temperature T (in oC) 95 94.3 93.5 92.8 92.1 91.3 90.6 89.9 89.2 88.5 87.9 3
10. Find the arc length of the curve y = 2(x − 1)2 between x = 1 and x = 3.
—END OF THE QUESTION PAPER. GOOD LUCK!—