HCMC UNIVERSITY OF TECHNOLOGY
AND EDUCATION
HIGH QUALITY TRAINING FACULTY
-------------------------
Question 1 (1 pt) Show that
2x 3
x2 1
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.
e x has at least one solution on
by using the root
location theorem.
Question 2 (2 pts) Evaluate the limit
x. e3 x 1
a. lim
b. lim( x 2
x 0 2cos x
x 0
2
1
e2 x ) x
Question 3 (2 pts)
a. Find the value of the constant k for which the following piecewise-defined function is
continuous everywhere.
sin x e 2 x 1
x 0
.
f x
x
m
x 0
b. Find f ' 0 .
Question 4 (1 pt) Let y be an implicit function of x satisfying:
(*)
sin x e 2 x 2 y 2 4 x 9
a/ Find
dy
dx
b/ Find the equation of the tangent line to the graph of equation (*) at the point P (0; 2) .
Question 5 (1 pt) Find the rectangle with largest area that
fits inside the graph of the parabola y x 2 below the line
y 4 , with the top side of the rectangle on the horizontal
line y 4 ; see the figure.
Question 6 (1 pt)
Water is poured into a conical container at the rate of 10 cm3/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
V
r 2h
. How fast is the water level rising when the water is
3
3 cm deep (at its deepest point)?
Question 7 (1 pt)
No.: BM1/QT-PĐBCL-RĐTV
Page: 1
Let f x 3x 2 4 x 5 . Find the average value of f on the interval 4,10 .
Question 8 (1 pt)
Find the particular solution of the separable differential equation satisfying the initial
condition:
dy
1
; y 1 4.
dx 2 xy (1 ln x )
Notice: Invigilators should not explain the questions on the exam papers.
Expected Learning Outcomes
[ELO 3.1]: Identify, analyze and use mathematical reasoning to
solve both problems involving theory and practical problems.
[ELO 2.1]: Present mathematical information using words,
statements, numbers, formulas, graphs and diagrams
[ELO 5.1]: Evaluate the limit of a function. Apply L’Hopital rule to
find limits involving infinity.
[ELO 5.2]: Find derivative and differential by using basic
derivatives and rules for derivatives.
[ELO 1.1, 1.3, 5.2]: Students are able to find basic limits and test
the continuity of a function. Students are able to find derivative and
differential.
[ELO 2.1, 1.2]: Students are able to use derivative to solve
problems relating to rates of change and optimization
[ELO 3.1, 5.4] : Apply important rules and theorems effectively,
such as the mean value. Students are able to apply theory to
evaluate indefinite and definite integrals.
[ELO 1.4, 5.4]: Students are able to solve basic differential
equations.
Questions
1
2,4
3
5,6
7
8
May 30, 2018
Head of foundation science group
No.: BM1/QT-PĐBCL-RĐTV
Page: 2
HCMC UNIVERSITY OF TECHNOLOGY
AND EDUCATION
SOLUTION OF CALCULUS 1 - MATH141601E
Date of exam: 31/05/2018
HIGH QUALITY TRAINING FACULTY
Question
Content
1
Denote
f ( x)
Score
0,25
2x 3 x
e which is contionous on the closed interval [0,1] .
2
x 1
0,25
f (0)
2.0 3 0
e
02 1
2 0
f (1)
2.1 3 1
e
12 1
0, 2182 0
0,25
Therefore, by the root location theorem there is at least one number c on 0,25
[0,1] for which f (c )
2
a
b
0,5
0,5
1
A lim( x 2
e2 x ) x
0
ln A ln lim( x 2
x
0
1
2x x
lim ln( x 2
e )
x
0
1
2x x
e )
lim
x
0
ln( x 2
0,5
e2 x )
x
2x
2 x 2e
x 0 x2
e2 x
ln A 2
A e2
LH
a
ex .
x. e3x 1
x
3
2x 3
x2 1
1 e3x 1 x2
1 e3x 1
lim .
.
lim .
.( 2)
0 2cos x 2
x 02
x cos x 1 x 0 2 x
1 e3x LH
3e3x
lim
lim
3
x 0
x 0
x
1
lim
x
0 and it follows that
lim
2
0,5
0,25
2x
1
0 : sin x e
is continuous
x
x
then the function is continuous every where if and only if f(x) is continuous
at x 0 . It means lim f ( x) f (0)
x
lim f ( x )
x
0
f (0)
x
0
sin x e 2 x 1 LH
cos x 2e 2 x
lim
0
x 0
x
1
lim
0,25
0,25
3
m
0,25
m 3
No.: BM1/QT-PĐBCL-RĐTV
Page: 1/1
b
4
a
sin x e 2 x
cos x 2e
b
2 y2
2x
0,5
4 x 9 , take derivative both sides
dy
4y
dx
dy
At P(0;2)
dx (0, 2)
4
0,5
4 cos x 2e 2 x
4y
dy
dx
4 cos 0 2.e 0
4.2
0,25
1
8
0,25
1
x 2
8
The equation of the tangent line to the curve is y
5
0,5
sin x e 2 x 1
3
f ( x) f (0)
sin x e 2 x 1 3 x
/
x
f (0) lim
lim
lim
x 0
x 0
x 0
x 0
x
x2
LH
cos x 2e 2 x 3 LH
sin x 4e 2 x
lim
lim
2
x 0
x 0
2x
2
Let A( x ) represent the area.
The lower right corner of the rectangle is at ( x, x 2 )
Then the area is A( x) 2 x (4 x 2 )
0,25
2 x3 8 x
We want the maximum value of A(x) when x is in [0; 4] [0; 2]
A/ ( x )
6x2 8
A/ ( x )
6x2 8 0
x
4
3
2 3
3
0,5
Testing this and the two endpoints, we have
4
A(0) 0 ; A(2) 0 ; A( )
3
0,25
32 3
9
The maximum area thus occurs when the rectangle has dimensions x
6
We have
dV
dt
V
0,25
10 cm3 / sec
That is, because of similar triangles,
r
h
10
30
r
h
3
3
1
3
0,25
r 2h
3
Take derivative both sides with respect to t ( Time is denoted by t )
No.: BM1/QT-PĐBCL-RĐTV
4
.
3
Page: 1/1
dV
dt
r 2 dh
.
3 dt
dV
dt
r 2 dh
.
3 dt
dh
dt
30
0,25
.12 dh
3 dt
10
9,55 cm / sec
0,25
7
8
1
10
AV
1
(3 x 2
64
dy
dx
1
2 xy (1 ln x )
4 x 5)dx 179
2 ydy
0,25
dx
x (1 ln x)
Take integral both sides:
2 y dy
y2
dx
x(1 ln x)
ln 1 ln x
We have y (1) 4
So the solution is
No.: BM1/QT-PĐBCL-RĐTV
0,25
c
0,25
42
y2
ln 1 ln1 c
ln 1 ln x
0,25
c 16
16 .
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