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Khánh Đặng Huy

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Test 1 School of Applied Mathematics and Informatics
Final Exam Analysis 1 - 20221
MI1114E, Time: 90 minutes
Question 1. Find the limit lim
x→+∞
√
x +
3
√
x +
4
√
x
√
2x + 1
.
Question 2. Find the limit lim
x→0
1 + x
2
cot
2
x
.
Question 3. Find the second derivative y = e
sin x
cos(sin x).
Question 4. Is the following function continuous at (0, 0)?
f (x, y) =
2x
2
−3y
2
x
2
+ y
2
if (x, y) = (0, 0),
0 if (x, y) = (0, 0).
Question 5. Find the first differential of u = x
y
z
.
Question 6. Calculate the integral
R
dx
(1 − x) (1 + x
2
)
.
Question 7. Calculate the integral
R
2
−2
dx
(2023
x
+ 1)(x
2
+ 4)
.
Question 8. Find the volume of the solid obtained by rotating
the region bouned by the curve x
2/3
+ y
2/3
= a
2/3
about y-
axis.
Question 9. Given f (x, y) = y
2
e
−(y
2
−x)
. Prove that for x =
t sin α, y = t cos α, the function f (x, y) → 0 as t → +∞ but
the limit lim
(x ,y)→(+∞,+∞)
f (x, y) doesn’t exist.
Question 10. Prove that the function f (x) = −
√
x is uni-
formly continuous on [0, +∞).
————————————–
One point for each question.
Test 2 School of Applied Mathematics and Informatics
Final Exam Analysis 1 - 20221
MI1114E, Time: 90 minutes
Question 1. Find the limit lim
x→+∞
q
x
p
x +
√
x
√
x + 1
.
Question 2. Find the limit lim
x→0
1 + tan x
1 + sin x
1
sin
3
x
.
Question 3. Find the second derivative y = e
cos x
sin(cos x).
Question 4. Is the following function continuous at (0, 0)?
f (x, y) =
x
2
y
x
4
+ y
2
nếu (x, y) = (0, 0),
0 nếu (x, y) = (0, 0).
Question 5. Find the first differential of u = z
y
x
.
Question 6. Calculate the integral
R
dx
(x + 1) (x
2
+ 1)
.
Question 7. Calculate the integral
R
3
−3
dx
(2023
x
+ 1)(x
2
+ 9)
.
Question 8. Find the volume of the solid obtained by rotating
the region bouned by the curve x
2/3
+ y
2/3
= a
2/3
about x-
axis.
Question 9. Given f (x, y) = x
2
e
−(x
2
−y)
. Prove that for x =
t cos α, y = t sin α, the function f (x, y) → 0 as t → +∞ but
the limit lim
(x ,y)→(+∞,+∞)
f (x, y) doesn’t exist.
Question 10. Prove that the function f (x) = −
√
x is uni-
formly continuous on [0, +∞).
————————————–
One point for each question.
1

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Test 1
School of Applied Mathematics and Informatics Test 2
School of Applied Mathematics and Informatics
Final Exam Analysis 1 - 20221
Final Exam Analysis 1 - 20221
MI1114E, Time: 90 minutes
MI1114E, Time: 90 minutes √ √ √ x + 3 x + 4 x q √ p
Question 1. Find the limit lim √ . x x + x x→+∞ 2x + 1
Question 1. Find the limit lim √ . x→+∞ x + 1 cot2 x
Question 2. Find the limit lim 1 + x2 . 1 x→0 1 + tan x
Question 2. Find the limit lim sin3 x .
Question 3. Find the second derivative y = esin x cos(sin x). x→0 1 + sin x
Question 4. Is the following function continuous at (0, 0)?
Question 3. Find the second derivative y = ecos x sin(cos x).  2x2 − 3y2
Question 4. Is the following function continuous at (0, 0)?  if (x, y) ̸= (0, 0), f (x, y) = x2 + y2  x2y 0 if (x, y) = (0, 0).  nếu (x, y) ̸= (0, 0), f (x, y) = x4 + y2 0 nếu (x, y) = (0, 0). y
Question 5. Find the first differential of u = x z . dx y
Question 6. Calculate the integral R .
Question 5. Find the first differential of u = z x . (1 − x)(1 + x2) dx dx
Question 6. Calculate the integral R .
Question 7. Calculate the integral R 2 . (x + 1)(x2 + 1) −2 (2023x + 1)(x2 + 4) dx
Question 8. Find the volume of the solid obtained by rotating
Question 7. Calculate the integral R 3 . −3 (2023x + 1)(x2 + 9)
the region bouned by the curve x2/3 + y2/3 = a2/3 about y-
Question 8. Find the volume of the solid obtained by rotating axis.
the region bouned by the curve x2/3 + y2/3 = a2/3 about x-
Question 9. Given f (x, y) = y2e−(y2−x). Prove that for x = axis.
t sin α, y = t cos α, the function f (x, y) → 0 as t → +∞ but
Question 9. Given f (x, y) = x2e−(x2−y). Prove that for x = the limit lim f (x, y) doesn’t exist. (x,y)→(+∞,+∞)
t cos α, y = t sin α, the function f (x, y) → 0 as t → +∞ but √ the limit lim f (x, y) doesn’t exist.
Question 10. Prove that the function f (x) = − x is uni- (x,y)→(+∞,+∞) √
formly continuous on [0, +∞).
Question 10. Prove that the function f (x) = − x is uni-
————————————–
formly continuous on [0, +∞). One point for each question.
————————————– One point for each question. 1

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