No.: BM1/QT-PĐBCL-RĐTV Page: 1
HCMC UNIVERSITY OF TECHNOLOGY
AND EDUCATION
HIGH QUALITY TRAINING FACULTY
-------------------------
FINAL EXAM, SEMESTER 2, 2017-2018
Subject: Calculus 1
Course code: MATH141601E
Number of pages: 02 pages.
Duration: 90 minutes.
Date of exam: 31/05/2018
Materials are allowed during the exam.
Question 1 (1 pt) Show that
2
2 3
1
x
x
e
x
+
=
+ has at least one solution on
ℝ
by using the root
location theorem.
Question 2 (2 pts)
Evaluate the limit
a.
(
)
3
0
. 1
lim
2cos 2
x
x
x e
x
→
−
−
b.
1
2 2
0
lim( )
x
x
x
x e
→
+
Question 3 (2 pts)
a. Find the value of the constant
k
for which the following piecewise-defined function is
continuous everywhere.
( )
2
sin 1
0
0
x
x e x
f x x
m x
+ −
≠
=
=
.
b. Find
(
)
' 0
f
.
Question 4 (1 pt)
Let
y
be an implicit function of
x
satisfying:
2 2
sin 2 4 9
x
x e y x
+ + = +
(*)
a/ Find
dy
dx
b/ Find the equation of the tangent line to the graph of equation (*) at the point
(0;2)
P
.
Question 5 (1 pt)
Find the rectangle with largest area that
fits inside the graph of the parabola
2
y x
=
below the line
4
y
=
, with the top side of the rectangle on the horizontal
line
4
y
=
; see the figure.
Question 6 (1 pt)
Water is poured into a conical container at the rate of 10 cm
3
/sec.
The cone points directly down, and it has a height of 30 cm and a
base radius of 10 cm; see figure below. Volume of the cone is
2
3
r h
V
π
=
. How fast is the water level rising when the water is
3 cm deep (at its deepest point)?
Question 7 (1 pt)
