Past Board Exam Problems in Differential Calculus
Engineering Mathematics (University of Nueva Caceres)
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Past Board Exam Problems
in Differential Calculus
1.
2.
3.
CE Board Exam November 1993
Find the second derivative of y by
implicit differentiation
from the
equation 4x2 + 8y2 = 36.
A. 64x2
C.
9.
4
1- 4x 2
B.
C.
-
x
4x  1
x
2(x + 1)2
x
3(x + 1
-
x
-
2x  13
x
(x + 1)3
x3
(x + 1)3
x2
9 3
y answer
4
32xy
16 3
y
9
-
CE Board Exam May 1998
Find the slope of the curve x2 + y2 –
6x + 10y + 5 = 0 at point (1, 0).
A. 1/5
B. 2/5
C. 1/4
D. 2
10. CE Board Exam May 1996
Find the slope 2of the tangent to the
curve y = 2x – x + x3 at (0, 2).
A. 1
B. 2
C. 3
D. 4
x
)2
D.
)
x +1
x
(x + 1)2 (x + 1)3
2
6.
8.
answer
CE Board Exam November 1995
The derivative with respect to x of
2cos2 (x2 + 2)
A. 2sin (x2 + 2) cos (x2 + 2)
B. -2sin (x2 + 2) cos (x2 + 2)
C. 8xsin (x2 + 2) cos (x2 + 2)
D. -8xsin (x2 + 2) cos (x2 + 2)
CE November 1997
What is the first derivative of y =
arcsin 3x?
11. CE Board Exam May 1996
Find the slope of the ellipse x2 + 4y2 –
10x – 16y + 5 = 0 at the point where y
= 2 + 80.5 and x = 7.
A. -0.1463
B. -0.1538
C. -0.1654
D. -0.1768
12. CE Board Exam May 1995
Find the equation of the line normal
to x2 + y2 = 25 at the point (4, 3).
A. 5x + 3y = 0
B. 3x – 4y = 0
C. 3x + 4y = 0
5x – 3y = 0
13. CE November 1998
Determine the slope of the curve x2 +
y2 – 6x – 4y – 21 = 0 at (0, 7).
A. 3/5
B. -2/5
C. -3/5
D.
answer
CE May 1999
Find the second derivative of y = x-2
at x = 2.
A. 96
B. 0.375
C. -0.25
D. -0.875
B.
CE Board Exam November 1996
3
A.
3
7.
D.
Find the derivative of
1 9x
3
2
1  9x 2
2
(
D.
3
1 9x 2
3

1  9x 2
D.
CE Board Exam May 1997
Find the derivative of arcos 4x.
-4
A.
answer
1- 16x 2
4
B.
1- 16x
-4
1- 4x 2

B.
C.
CE Board Exam November 1994
What is the derivative with respect to
x of (x + 1)3 – x3?
A. 3x + 6
B. 3x – 3
C. 6x – 3
D. 6x + 3
D.
5.

CE Board Exam November 1997
x2 - 1
Evaluate Lim
2
x 1 x  3x - 4
A. 1/5
B. 2/5
C. 3/5
D. 4/5
C.
4.
A.
2/5
14. CE May 1998
Find the slope of the line whose
parametric equations are x = 4t + 6
and y = t – 1.
A. -4
B. 1/4
C. 4
D. -1/4
15. CE November 1999
Find the slope of the curve y = 64(4 +
x)1/2 at (0, 12).
A. 0.67
B. 1.5
C. 1.33
D. 0.75
16. CE May 1999
Find the angle that the line 2y – 9x –
18 = 0 makes with the x-axis.
A. 74.77°
B. 4.5°
C. 47.77°
D. 77.47°
17. CE November 1998
What is the radius of curvature at
point (1, 2) of the curve 4x – y2 = 0?
A. 6.21
B. 5.21
C. 5.66
D. 6.66
18. CE November 1997
Find the radius of curvature of the
curve y2 - 4x = 0 at point (4, 4).
A. 24.4
B. 25.4
C. 23.4
D. 22.4
19. CE November 1999
Find the radius of curvature of the
curve x = y3 at (1, 1).
A. 4.72
B. 3.28
C. 4.67
D. 5.27
20. CE November
The chords of the ellipse 64x2 + 25y2
= 1600 having equal slopes of 1/5 are
bisected by its diameter. Determine
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the equation of the diameter of the
ellipse?
A. 5x – 64y = 0
B. 64x – 5y = 0
C. 5x + 64y = 0
D. 64x + 5y = 0
21. CE Board Exam May 1995
A wall “h” meters high is 2 m away
from the building. The shortest ladder
that can reach the building with one
end resting on the ground outside the
wall is 6 m. How high is the wall in
meters?
A. 2.34
B. 2.24
C. 2.44
D. 2.14
22. CE Board Exam May 1997
Find the minimum amount of tin
sheet that can be made into a closed
cylinder having a volume of 108 cu.
inches in square inches.
A. 125.50
B. 127.50
C. 129.50
D. 123.50
23. CE Board Exam November 1996
A Norman window is in the shape of
a rectangle surmounted by a semicircle. What is the ratio of the width of
the rectangle to the total height so
that it will yield a window admitting
the most light for a given perimeter?
A. 1
B. 1/2
C. 2
D. 2/3
24. CE Board Exam May 1998
Determine the diameter of a closed
cylindrical tank having a volume of
11.3 cu. m to obtain the minimum
surface area.
A. 1.22
B. 1.64
C. 2.44
D. 2.68
25. CE Board Exam November 1998
Water is pouring into a conical vessel
15 cm deep and having a radius of
3.75 cm across the top if the rate at
which the water rises is 2 cm/sec,
how fast is the water flowing into the
conical vessel when the water is 4
cm deep?
A. 2.37 m3/sec
B. 5.73 m3/sec
C. 6.28 m3/sec
D. 4.57 m3/sec
26. CE Board Exam May 1997
Car A moves due east at 30 kph at
the same instant car B is moving S
30° E, with a speed of 60 kph. If the
distance from A to B is 30 km, find
how fast is the distance between
them separating after one hour.
A. 36 kph
B. 38 kph
C. 40 kph
D. 45 kph
27. CE Board Exam November 1996
A car starting at 12:00 noon travels
west at a speed of 30 kph. Another
car starting from rest starting from the
same point at 2:00 pm travels north
at 45 kph. Find how fast (in kph) the
two are separating at 4:00 pm?
A. 49
B. 51
C. 53
D. 55
28. CE Board Exam May 1996
Two railroad tracks are perpendicular
to each other. At 12:00 pm there is a
train at each track approaching the
crossing at 50 kph, one being 100 km
and the other 150 km away from the
crossing. How fast in kph is the
distance between the two trains
changing at 4:00 pm?
A. 67.08
B. 68.08
C. 69.08
D. 70.08
29. CE Board Exam May 1998
Water is running into a hemispherical
bowl having a radius of 10 cm at a
constant rate of 3 cm3/min. When the
water is x cm deep, the water level is
rising at the rate of 0.0149 cm/min.
What is the value of x?
A. 3
B. 2
C. 4
D. 5
30. CE May 1999
The number of newspaper copies
distributed is given by C = 50t2 – 200t
+ 10000, where t is in years. Find the
minimum
number
of
copies
distributed from 1995 to 2002
A. 9850
B. 9800
C. 10200
D. 7500
31. CE May 1999
A rectangular box having a square
base and open at the top is to have a
capacity of 16823 cc. Find the height
of the box to use the least amount of
material.
A. 16.14 cm
B. 32.28 cm
C. 18.41 cm
D. 28.74 cm
32. CE May 1998, November 1999
The volume of a closed cylindrical
tank is 11.3 cubic meters. If the total
surface is a minimum, what is its
base radius, in m?
A. 1.44
B. 1.88
C. 1.22
D. 1.66
33. CE May 1998
An object moves along a straight line
such that, after t minutes, its distance
from its starting point is D = 20t + 5/(t
+ 1) meters. At what speed, in
m/minute will it be moving at the end
of 4 minutes?
A. 39.8
B. 49.8
C. 29.8
D. 19.8
34. CE May 1996
The speed of the traffic flowing past a
certain downtown exit between the
hours of 1:00 pm and 6:00 pm is
approximately V = t3 – 10.5t2 + 30t +
20 miles per hour, where t = number
of hours past noon. What is the
fastest speed of the traffic between
1:00 pm and 6:00 pm in mph?
A. 50
B. 46
C. 32.5
D. 52
35. CE May 1997
A car drives east from point A at 30
kph. Another car B starting from B at
the same time drives S 30° W toward
A at 60 kph. B is 30 km away from A.
How fast in kph is the distance
between the two cars changing after
one hour?
A. 76.94 kph
B. 78.94 kph
C. 75.94 kph
D. 77.94 kph
36. CE November 1998
There is a constant inflow of a liquid
into a conical vessel 15 ft deep and
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7.5 feet in diameter at the top. Water
is rising at the rate of 2 feet per
minute when the water is 4 feet deep.
What is the rate of inflow in cu. Ft per
minute?
A. 8.14
B. 7.46
C. 9.33
D. 6.28
37. CE May 2003
What is the radius of curvature of the
curve y2 = 16x = 0 at the point (4, 8)?
A. -0.044
B. -0.088
C. -0.066
D. -0.033
38. CE November 2002
Suppose that x years after founding
in 1975, a certain employee
association has a membership of f(x)
= 100(2x3 – 45x2 + 264x), at what time
between 1975 and 1989 was the
membership smallest?
A. 1983
B. 1985
C. 1984
D. 1986
39. CE November 2002
A 3 meter long steel pipe has its
upper end leaning against a vertical
wall and lower end on a level ground.
The lower end moves away at a
constant rate of 2 cm/s. How fast is
the upper end moving down, in cm/s,
when the lower end is 2 m from the
wall?
A. 1.81
B. 1.66
C. 1.79
D. 1.98
40. CE May 2002
A particle moves according to the
parametric equations:
y = 2t2 and x = t3
where x and y are displacement (in
meters) in x and y direction,
respectively and t is time in seconds.
Determine the acceleration of the
body after t = 3 seconds.
A. 12.85 m/s2
B. 18.44 m/s2
C. 21.47 m/s2
D. 5.21 m/s2
41. CE May 2002
Determine the shortest distance from
point (4, 2) to the parabola y2 = 8x.
A. 2.83
B. 3.54
C.
D.
Differentiate y = sec (x2 + 2)
A. 2x cos (x2 + 2)
B. -cos (x2 + 2) cot (x2 + 2)
C. 2x sec (x2 + 2) tan (x2 + 2)
D. cos (x2 + 2)
2.41
6.32
42. CE November 2001
Water flows into a tank having the
form of a frustum of a right circular
cone. The tank is 4 m tall with upper
radius of 1.5 m and the lower radius
of 1 m. When the water in the tank is
1.2 m deep, the surface rises at the
rate of 0.012 m/s. Calculate the
discharge of water flowing into the
tank in m3/s.
A. 0.02
B. 0.05
C. 0.08
D. 0.12
49. EE Board Exam October 1997
Differentiate y = log(x2+1)2
A. 4x(x2 + 1)
4x log10 e
B.
answer
(x 2  1)
C. log e(x)(x2 + 1)
D. 2x(x2 + 1)
50. EE Board Exam October 1997
Differentiate (x2 + 2)1/2
A.
43. CE November 2003
The motion of a particle is defined by
the parametric equation x = t3 and y =
2t3. Determine the velocity when t =
2.
A. 14.42
B. 16.25
C. 12.74
D. 18.63
B.
C.
D.
B.
C.
D.
C.
D.
3x4  2x2  7
5x 3  x  3
Undefined
3/5
infinity
0
47. EE Board Exam October 1997
Differentiate y = ex cos x2
A. -ex sin x2
B. ex (cos x2 – 2x sin x2)
C. ex cos x2 – 2x sin x2
D. -2xex sin x
48. EE Board Exam October 1997
2x2 + 2x
3
2(x + 2)
x5/2 + x1/2
1
2
53. EE Board Exam October 1997
If y = 4cos x + sin 2x, what is the
slope of the curve when x = 2?
A. -2.21
B. -4.94
C. -3.25
D. -2.21
46. EE Board Exam October 1994
A.
B.
C.
D.
(x2 + 2)3/2
52. EE Board Exam April 1995
Find y’ if y = arcsin (cos x).
A. -1
B. -2
45. EE Board Exam April 1993
1 cos x
Evaluate Lim
x0
x2
A. 0
B. 1/2
C. 2
D. -1/2
x
(x 2  2)1/ 2 answer
2x
(x 2  2)1/ 2
51. EE Board Exam October 1997
If y = (t2 + 2)2 and t = x1/2, determine
dy/dx.
A. 3/2
44. CE November 2003
The sum of two numbers is K. The
product of one by the cube of the
other is to be a minimum. Determine
one the numbers.
A. 3K/4
B. 3K/8
C. 3K/2
D. 3K/7
Evaluate Lim
(x 2  1)1/ 2
2
x
54. EE Board Exam April 1997
Locate the points of inflection of the
curve y = f(x) = x2ex.

A.
2  3
B.
2
C.
2  2 answer
D.
2
55. EE Board Exam April 1990
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The sum of two positive numbers is
50. What are the numbers if their
product is to be the largest possible.
A. 24 and 26
B. 28 and 22
C. 25 and 25
D. 20 and 30
56. EE Board Exam March 1998
A triangle has variable sides x, y, z
subject to the constraint such that the
perimeter P is fixed to 18 cm. What is
the maximum possible area for the
triangle?
A. 15.59 cm2
B. 18.71 cm2
C. 17.15 cm2
D. 14.03 cm2
57. EE Board Exam October 1997
A farmer has enough money to build
only 100 meters of fence. What are
the dimensions of the field he can
enclose this maximum area?
A. 25 m by 25 m
B. 15 m by 35 m
C. 20 m by 30 m
D. 22.5 m by 27.5 m
58. EE Board Exam April 1997
The cost of fuel in running a
locomotive is proportional to the
square of the speed and is $25 per
hour for a speed of 25 miles per hour.
Other costs amount to $ 100 per
hour, regardless of the speed. What
is the speed which will make the cost
per mile a minimum?
A. 40
B. 55
C. 50
D. 45
59. EE Board Exam April 1997
A poster is to contain 300 m2 of
printed matter with margins of 10 cm
at the top and bottom and 5 cm at
each
side.
Find
the
over-all
dimensions, if the total area of the
poster is a minimum.
A. 27.76 cm, 47.8 cm
B. 20.45 cm, 35.6 cm
C. 22.24 cm, 44.5 cm
D. 25.55 cm, 46.7 cm
60. EE Board Exam March 1998
A fencing is limited to 20 ft in length.
What is the maximum rectangular
area that can be fenced in using two
perpendicular corner sides of an
existing wall?
A. 120
B. 100
C.
D.
140
190
61. EE Board Exam October 1992
The cost per hour of running a motor
boat is proportional to the cube of the
speed. At what speed will the boat
run against a current of 8 kph in order
to go a distance most economically?
A. 10 kph
B. 13 kph
C. 11 kph
D. 12 kph
62. EE Board Exam October 1993
A boatman is at A which is 4.5 km
from the nearest point B on a straight
shore BM. He wishes to reach in
minimum time a point C situated on
the shore 9 km from B. How far from
C should he land if he can row at the
rate of 6 kph and can walk at the rate
of 7.5 kph?
A. 4.15 km
B. 3.0 km
C. 3.25 km
D. 4.0 km
A.
B.
C.
D.
0.64
0.56
0.75
0.45
m/min
m/min
m/min
m/min
66. EE Board Exam October 1993
A standard cell has an emf “E” of 1.2
volts. If the resistance “R” of the
circuit is increasing at a rate of 0.03
ohm/sec, at what rate is the current
“I” changing at the instant when the
resistance is 6 ohms? Assume
Ohm’s law E = IR.
A. -0.002 amp/sec
B. 0.004 amp/sec
C. -0.001 amp/sec
D. 0.003 amp/sec
67. ECE Board Exam November 1991
(
)
2
Evaluate the limit Lim x + 3x - 4
x →4
A.
B.
C.
D.
24
26
28
30
68. ECE Board Exam November 1994
πx
63. EE Board Exam October 1993
At any distance x from the source of
light, the intensity of illumination
varies directly as the intensity of the
source and inversely as the square of
x. Suppose that there is a light at A
and another at B having an intensity
8 times that of A. The distance AB is
4 m. At what point from A on the line
AB will the intensity of illumination be
least?
A. 2.15 m
B. 1.33 m
C. 1.50 m
D. 1.92 m
64. EE Board Exam April 1997
The coordinates (x, y) in feet of a
moving particle P is given by x = cos t
– 1 and y = 2sin t + 1, where t is the
time in seconds. At what extreme
rates in fps is P moving along the
curve?
A. 3 and 2
B. 3 and 1
C. 2 and 0.5
D. 2 and 1
65. EE Board Exam October 1993
Water is flowing into a conical cistern
at the rate of 8 m3/min. If the height of
the inverted cone is 12 m and the
radius of its circular opening is 6 m.
How fast is the water level rising
when the water is 4 m deep?
Evaluate Lim(2 - x)tan 2
x →1
A.
B.
C.
D.
e2π
e2/π
0
∞
69. ECE Board Exam April 1998
x- 4
Evaluate lim
x® 4 x2
A.
B.
C.
D.
- x - 12
undefined
0
infinity
1/7
70. ECE Board Exam April 1993
2
Evaluate lim x - 4
x® 2 x - 2
A. 0
B. 2
C. 4
D. 6
71. ECE November 1997
Evaluate the limit (ln x)/x as x
approaches positive infinity.
A. 1
B. 0
C. e
D. infinity
72. ECE Board Exam November 1991
Downloaded by Carl Vincent P. Taboada (lract090@gmail.com)
Differentiate the equation y =
x 2 + 2x
A.
B.
C.
D.
(x + 1)2
x2
x +1
answer
x
x +1
2x
2x 2
x +1
73. ECE November1997
Given the equation y = (elnx)2, find y’
A. lnx
B. 2( ln x) /x
C. 2x
D. 2eln x
74. ECE March 1996
The derivative of ln (cos x) is
A. sec x
B. -sec x
C. -tan x
D. tan x
vertex, the slope of the tangent line is
zero.
A. (2, -3)
B. (3, -2)
can be inscribed in the cone to the
volume of the cone?
A. 44%
B. 46%
C.
D.
C.
D.
(-1, -3)
(-2, 3)
80. ECE Board Exam April 1999
Find the equation of the line normal
to x2 + y2 = 5 at the point (2, 1).
A. y = 2x
B. x = 2y
C. 2x+ 3y = 3
D. x + y = 1
81. ECE Board Exam November 1991
In the curve y = 2 + 12x – x3, find the
critical points.
A. (2, 18) and (-2, -14)
B. (-2, 18) and (2, -14)
C. (2, 18) and (2, -14)
D. (-2, 18) and (-2, 14)
82. ECE Board Exam November 1996
Find the radius of curvature of a
parabola y2 – 4x = 0 at point (4, 4).
A. 22.36 units
B. 25.78 units
C. 20.33 units
D. 15.42 units
75. ECE November 1997
If y = x(ln x), find d2y/dx2.
A. 1/x2
B. -1/x
C. 1/x
D. -1/x2
76. ECE Board Exam November 1991
Find the slope of the line tangent to
the curve y = x3 – 2x + 1 at x = 1.
A. 1
B. ½ asss
C. 1/3
D. 1/4
77. ECE Board Exam November 1991
Give the slope of the curve at the
point (1, 1): y = x3/4 – 2x + 1.
A. 1/4
B. -1/4
C. 1 1/4
D. -1 1/4
78. ECE November 1998
Find the slope of x2y = 8 at the point
(2, 2).
A. 2
B. -1
C. -1/2
D. -2
79. ECE Board Exam April 1999
Find the coordinates of the vertex of
the parabola y = x2 – 4x + 1 by
making use of the fact that at the
83. ECE Board Exam November 1996
Find the radius of curvature at any
point in the curve y + ln cos x = 0.
A. cos x
B. 1.5707
C. sec x
D. 1
84. ECE Board Exam April 1999
Find the minimum distance from the
point (4, 2) to the parabola y2 = 8x.
A.
4
B.
2
answer
C.
D.
2
85. ECE April 1998
Find the point in the parabola y2 = 4x
at which the rate of change of the
ordinate and abscissa are equal.
A. (1, 2)
B. (2, 1)
C. (4, 4)
D. (-1, 4)
56%
65%
87. ECE Board Exam April 1998
A statue 3 m high is standing on a
base of 4 m high. If an observer’s eye
is 1.5 m above the ground, how far
should he stand from the base in
order that the angle subtended by the
statue is a maximum?
A. 3.41 m
B. 3.51 m
C. 3.71 m
D. 4.41 m
88. ECE Board Exam November 1991
A balloon is released from the ground
100 meters from an observer. The
balloon rises directly upward at the
rate of 4 meters per second. How fast
is the distance between them
changing after 1 second?
A. 1.68 m/sec
B. 1.36 m/sec
C. 1.55 m/sec
D. 1.49 m/sec
89. ECE Board Exam April 1998
A balloon is rising vertically over a
point A on the ground at the rate of
15 ft/sec. A point B on the ground is
level with and 30 ft from A. When the
balloon is 40 ft from A, at what rate is
its distance from B changing?
A. 13 fps
B. 15 fps
C. 12 fps
D. 10 fps
90. ECE Board Exam November 1998
What is the allowable error in
measuring the edge of the cube that
is intended to hold 8 cu. m, if the
error of the computed volume is not
to exceed 0.03 cu. m?
A. 0.002
B. 0.003
C. 0.0025
D. 0.001
91. ECE Board Exam November 1997,
November 1999
86. ECE Board Exam November 1996
Given a cone of diameter x and
altitude of h. What percent is the
volume of the largest cylinder which
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If y = x ln x, find
A.
B.
C.
D.
-1/x
-1/x2
1/x2
1/x
d2y
dx2
92. ECE Board Exam April 1999
The depth of water in a cylindrical
tank 4 m in diameter is increasing at
the rate of 0.7 m/min. Find the rate at
which the water is flowing into the
tank.
A. 2.5 m3/min
B. 1.5 m3/min
C. 6.4 m3/min
D. 8.8 m3/min
93. ECE Board Exam November 1999
Two posts, one 8 m high and the
other 12 m high are 15 m apart. If the
posts are supported by a cable
running from the top if the first post to
a stake on the ground and then back
to the top of the second post, find the
distance to the lower post to the
stake to use minimum amount of
wire.
A. 6 m
B. 8 m
C. 9 m
D. 4 m
94. ECE Board Exam April 2000
Find the approximate increase by the
use of differentials, in the volume of
the sphere if the radius increases
from 2 to 2.05 in one second.
A. 2.12
B. 2.51
C. 2.86
D. 2.25
95. ECE Board Exam April 2000, April
1999
What is the area of the largest
rectangle that can be inscribed in a
semi-circle of radius 10?
2 50
A.
B.
C.
100
1000
D.
50
96. ECE Board Exam April 2000
If ln(ln y) + ln y = ln x, find y’.
A. x/(x + y)
B. x/(x – y)
C. y/(x + y)
D. y/(x - y)
97. ECE Board Exam April 2000
The volume of the sphere is
increasing at the rate of 6 cm3/hr. At
what rate is its surface area
increasing (in cm2/hr) when the
radius is 50 cm?
A. 0.50
B. 0.30
C.
D.
= 18t + 9t2. What is the velocity at t =
3?
A. 36
B. 18
C. 72
D. 54
0.40
0.24
98. ECE Board Exam April 2000,
November 2001
Water is running out a conical funnel
at the rate of 1 cu. in per second. If
the radius of the base of the funnel is
4 in and the altitude is 8 in., find the
rate at which the water level is
dropping when it is 2 in. from the top.
A. -3/2 pi in/s
B. 2/3 pi in/s
C. -4/9 pi in/s
D. -1/9 pi in/s
99. ECE Board Exam November 2000
If y = 2x + sin 2x, find x if y’= 0.
A. π/2
B. π/4
C. 2π/3
D. 3π/2
100. ECE Board Exam November 2000
The equation of the line tangent to
the curve y = x + 5/x at point P(1, 3)
is
A. 4x – y + 7 = 0
B. x + 4y – 7 = 0
C. 4x + y – 7 = 0
D. x – 4y + 7 = 0
101. ECE Board Exam November 2000
If y = arctan (ln x), find dy/dx at x =
1/e.
A. e
B. e/2
C. e/3
D. e^2
105. ECE Board Exam November 2001
Find the height of a right circular
cylinder of maximum volume, which
can be inscribed in a sphere of radius
10 cm.
A. 14.12 cm
B. 15.11 cm
C. 12.81 cm
D. 11.55 cm
106. ECE Board Exam November 2001
What is the second derivative of a
function y = 5x3 + 2x + 1?
A. 25x
B. 30x
C. 18
D. 30
107. ECE Board Exam April 1999, April
2002
Find the minimum distance from the
point (4,2) to the parabola y2 = 8x.
A. 4
B.
D.
answer
3
2
108. ECE Board Exam November 2002,
November 2004
A statue 3.2 m high stands on a
pedestal such that its foot is 0.4 m
above an observer’s eye level. How
far from the statue must the observer
stand in order that the angle
subtended by the statue will be a
maximum?
A. 1.1 m
B. 1.5 m
C. 1.2 m
D. 1.4 m
102. ECE Board Exam November 2000
Find the change in y = 2x – 3 if x
changes from 3.3 to 3.5.
A. 0.4
B. 0.2
C. 0.5
D. 0.3
103. ECE Board Exam April 2001
The radius of a sphere is r inches at
time t seconds. Find the radius when
the rates of increase of the surface
area and the radius are numerically
equal.
A. 1/8π in.
B. 1/4π in.
C. 2π in.
D. π2 in.
2
C.
109. ECE Board Exam November 2002
104. ECE Board Exam November 2001
The distance of a body travels as a
function of time and is defined by x(t)
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A person in a rowboat is 3 km from a
point P on a straight shore while his
destination is 5 km directly east of
point P. If he is able to row 4 km per
hour and walk 5 km per hour, how far
from the destination must he land on
the shore in order to reach his
destination in the shortest possible
time?
A. 1 km
B. 2.5 km
C. 3 km
D. 2 km
110. ECE Board Exam November 2002
What is the slope of the curve y = 1 +
x2 at the point where y = 10?
A. 8
B. 3
C. 9
D. 6
111. ECE Board Exam November 2002
Given the equation: 2y3 = 3x2 – 5.
Determine the slope of the line
tangent at (4, 1).
A. 4
B. 3
C. 1/4
D. 1
112. ECE Board Exam April 2002
What is the maximum area of a
rectangle that can be inscribed in a
right triangle with base of 8 cm and a
height of 6 cm?
A. 12 sq cm
B. 48 sq cm
C. 24 sq cm
D. 50 sq cm
113. ECE Board Exam April 2003
2
B.
3 2
C.
8 2 answer
D.
5 2
Determine the slope of the tangents
to the parabola y = -x2 + 5x – 6 at the
points of intersection with the x-axis.
A. 2
B. -4
C. 1
D. -2
115. ECE Board Exam April 2003
A drop of ink is placed on a piece of
paper and causes a circular blot that
increases in area at the rate of 1 sq
mm/sec. At what rate does the radius
of the bolt increase when its area is 1
sq mm?
1/
π2/2
/ 2π answer
C.
D.
π/
116. ECE Board Exam April 2003
-1
2pi
0
-2
123. ECE Board Exam November 2004
A conical vessel 1 cm deep and with
a radius of 6 cm at the top, is being
filled with water. If the rate at which
the water rises is 2 cm/sec, how fast
is the volume increasing when the
water is 4 cm deep?
A. 3 pi
B. 4 pi
C. 8 pi
D. 16 pi
117. ECE Board Exam November 2003
Determine the equation of the line
tangent to the parabola y = x2 at the
point (1, 1).
A. y = 2x + 1
B. y = 2 – x
C. y = 2x – 1
D. y = 2 + x
118. ECE Board Exam November 2003
A fisherman on a wharf 1.2 meters
above the level of still water is pulling
a rope tied to a boat at the rate of 2
meters per minute. How fast is the
boat approaching the wharf when
there are two meters of rope out?
A. 2.5 m/min
B. 1.25 m/min
C. 2.0 m/min
D. 3.0 m/min
124. ECE Board Exam November 2004
A customer is using a straw to drink
iced tea from a right circular glass at
the rate of 6 cubic cm per minute. If
the height of the glass is 12 cm and
the diameter is 6 cm, how fast is the
level of the iced tea decreasing at a
constant rate in cm per min?
A. 0.212
B. 1.570
C. 0.318
D. 0.747
What is the second derivative of y
with respect to w in the following
equation: y = (3w2 – 4)(3w2 + 4)?
A. 36w3
B. 9w4
C. 9w16
D. 108w2
125. ECE Board Exam November 2003
A condominium is to be constructed
in a rectangular lot with a perimeter
of 800 m. What is the largest area
that can be enclosed by fencing the
perimeter?
A. 5 hectares
B. 4 hectares
C. 6 hectares
D. 3.5 hectares
120. ECE Board Exam November 2003
114. ECE Board Exam April 2003
A.
B.
A.
B.
C.
D.
119. ECE Board Exam November 2003
Solve for the radius of a right circular
cone of maximum volume which can
be inscribed in a sphere of radius 12
cm.
A.
Solve for dy/dx if x = 2 + t and y = 1 +
t2.
A. 2x
B. t
C. 0
D. 2t
A stone is thrown into still water and
causes concentric circular ripples.
The radius of the ripples increases at
the rate of 12 inches/sec. At what
rate does the area of the ripples
increases (in sq inch per sec) when
its radius is 3 inches?
A. 402.55
B. 275.60
C. 226.19
D. 390.50
126. ECE Board Exam April 2005
The cost of a product is a function of
the quantity q of the product: c(q) = q2
– 2000q + 100. What should be the
quantity for which the cost is a
minimum?
A. 2500
B. 1000
C. 2000
D. 1500
121. ECE Board Exam November 2004
In how many equal parts can a wire,
50 cm long be cut so that the product
of its parts is a maximum?
A. 15
B. 19
C. 13
D. 20
127. ECE November 1996
122. ECE Board Exam November 2003
æsin x3 ÷
ö
ç
÷
Evaluate lim
2
x® 0ççèsin x ÷
ø
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A rectangular field is to be fenced
into four equal parts. What is the size
of the largest field that can be fenced
this way with a fencing length of 1500
feet if the division is to be paralleled
to one side?
A. 65,200
B. 62,500
C. 64,500
D. 63,500
128. ECE November 1997
Three sides of a trapezoid are each 8
cm long. How long is the 4th side,
when the area of the trapezoid has
the greatest value?
A. 16 cm
B. 15 cm
C. 12 cm
D. 10 cm
129. ECE March 1996, November 1996
The altitude of a cylinder of maximum
volume that can be inscribed in a
right circular cone of radius r and
height h is
A. h/3
B. 2h/3
C. 3h/2
D. h/4
130. ECE November 1995
The height of a right circular cylinder
is 50 inches and decreases at the
rate of 4 inches per second, while the
radius of the base is 20 inches and
increases at the rate of one inch per
second. At what rate is the volume
changing?
A. 11310 cu. m/sec
B. 1275 cu. m/sec
C. 11130 cu. m/sec
D. 1257 cu. m/sec
131. ECE November 1995, March 1996
A point on the curve where the
second derivative of a function is
equal to zero is called
A. maxima
B. minima
C. point of inflection
D. point of intersection
132. ECE November 1995
The point on the curve where the first
derivate of a function is zero and the
second derivative is positive is called
A. maxima
B. minima
C. point of inflection
D. point of intersection
A.
B.
C.
D.
At the minimum point, the slope of
the tangent line is
A. negative
B. infinity
C. positive
D. zero
134. ECE November 1996
B.
C.
135. ECE April 1998
Point of the derivatives, which do not
exist (and so equals zero) are called
A. stationary points
B. maximum points
C. maximum and minimum points
D. minimum point
D.
If the second derivative of the
equation of a curve is equal to the
negative of the equation of the same
curve, the curve is:
A. a cissoid
B. a paraboloid
C. a sinusoid
D. an exponential
A.
B.
C.
D.
Lim
x-4
A.
B.
C.
D.
following
Find the derivative of the function 2x2
+ 8x + 9 with respect to x.
A. DF(x) = 4x - 8
B. DF(x) = 2x + 9
C. DF(x) = 2x + 8
D. DF(x) = 4x + 8
limit:
144. ME April 1998
Differentiate ax2 + b to the 1/2 power.
A. -2ax
B. 2ax
C. 2ax+ b
D. ax + 2b
1
0
2
infinite
139. ME Board Exam April 1997
What is the first derivative of the
expression (xy)x = e?
A. 0
x
B.
y
(1+ ln xy)
answer
C. - y
x
-y
145. ME April 1997
If y
A.
B.
C.
D.
Given the function f(x) = x3 – 5x + 2,
find the value of the first derivative at
x = 2, f’(2).
A. 7
B. 3x2 - 5
C. 2
D. 8
(1- ln xy)
x
Find the derivative with respect to x
the function
- 2x2
= cos x, what is dy/dx?
sec x
-sec x
sin x
-sin x
146. ME October 1997
140. ME Board Exam April 1998
A.
3x
If a is a simple constant, what is the
derivative of y = xa?
A. axa-1
B. (a – 1)x
C. xa-1
D. ax
138. ME Board Exam October 1997
the
2 - 3x2
143. ME April 1996
0
1
8
16
Compute
x+ 4
lim
x® ¥ x - 4
- 3x2
142. ME April 1996
x 2 - 16
x 4
answer
Find the derivative of (x + 5)/(x2 – 1)
with respect to x.
A. DF(x) = (-x2 – 10x – 1)/(x2 - 1)2
B. DF(x) = (-x2 + 10x – 1)/(x2 - 1)2
C. DF(x) = (x2 – 10x – 1)/(x2 - 1)2
D. DF(x) = (-x2 – 10x + 1)/(x2 + 1)2
137. ME Board Exam April 1998
Evaluate
-3x
141. ME April 1996
136. ECE November 1997
D.
133. ECE November 1996
f”(a) = is not equal to zero
f”(a) = 0
f”(a) > 0
f”(a) < 0
147. ME Board Exam April 1998
2 - 3x2
At the inflection point of y = f(x)
where x = a,
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Find the partial derivative with
respect to x of the function xy2 – 5y +
6.
A. y2 – 5
B. y2
C.
D.
given below? P is profit and x is unit
of production.
P = 200,000 – x – [1.1/(x + 1)]8
A. 285,000
B. 200,000
C. 250,000
D. 305,000
xy – 5y
2xy
148. ME Board Exam October 1997
Find the second derivative of x3 – 5x2
+x=0
A. 10x - 5
B. 6x - 10
C. 3x + 10
D. 3x2 – 5x
155. ME Board Exam October 1996
Water is pouring into swimming pool.
After t hours, there are t + t1/2 gallons
in the pool. At what rate is the water
pouring into the pool when t = 9
hours?
A. 7/6 gph
B. 8/7 gph
C. 6/5 gph
D. 5/4 gph
149. ME Board Exam April 1998
Given the function f(x) = x to the 3rd
power -6x + 2. Find the first derivative
at x = 2.
A. 6
B. 7
C. 3x2 - 5
D. 8
156. ME October 1997
A function is given below, what x
value maximizes y?
y2 + y + x2 – 2x = 5
A. 2.23
B. -1
C. 5
D. 1
150. ME April 1996
Find the slope of the line tangent to
the curve y = x3 – 2x + 1 at x = 1.
A. 1
B. 1/2
C. 1/3
D. 1/4
157. ME April 1998
151. ME April 1996
If y = x to the 3rd power – 3x. Find the
maximum value of y.
A. 0
B. -1
C. 1
D. 2
Find the slope of the tangent to a
parabola y = x2 at a point on a curve
where x = 1/2.
A. 0
B. 1
C. 1/4
D. -1/2
158. ME April 1998
As x increases uniformly at a rate of
0.002 feet per second, at what rate is
the expression (1 + x) to the 3rd
power increasing when x becomes 8
feet?
A. 430 cfs
B. 0.300 cfs
C. 0.486 cfs
D. 0.346 cfs
152. ME Board Exam April 1998
A box is to be constructed from a
piece of zinc 20 sq. in. by cutting
equal squares from each corner and
turning up the zinc to form the side.
What is the volume of the largest box
that can be so constructed?
A. 599.95 cu. inches
B. 592.58 cu. inches
C. 579.50 cu. inches
D. 622.49 cu. inches
159. ME April 1998
The distance a body travels is a
function of time and is given by x(t) =
16t + 8t2. Find its velocity at t = 3.
A. 64
B. 56
C. 54
D. 44
153. ME Board Exam April 1996
The cost C of a product is a function
of the quantity x of the product: C(x)
= x2 – 4000x + 50. Find the quantity
for which the cost is a minimum.
A. 1000
B. 1500
C. 2000
D. 3000
154. ME Board Exam October 1996
What is the maximum profit when the
profit versus production function is as
160.
A.
B.
C.
D.
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