MATHEMATICS DIFFERENTIAL CALCULUS CE LICENSURE EXAMINATION PROBLEMS DIFFERENTIAL CALCULUS FIRST ORDER DERIVATIVE 1.
2.
3.
4.
Find y’ if y = sinh x. (M94 M 16) a. csch x c. b. cosh x d.
log u
b.
u ln a
7.
d.
Find the derivative with respect to x of 2 cos2 (x2 + 2). (N95 M 3) a. 4 sin (x2 + 2) cos (x2 + 2) b. -4 sin (x2 + 2) cos (x2 + 2) c. 8x sin (x2 + 2) cos (x2 + 2) d. -8x sin (x2 + 2) cos (x2 + 2)
10.
Find the derivative of: (N96 M 12) (x + 1)3 x
loga e du u dx du log a dx
a. b. c.
Find y’ if y = arcsin x. (M94 M 23)
b.
6.
c.
9.
Find the derivative h with respect to u of h = 2u. (M94 M 22) a. 2u c. 22u ln b. 2u ln d. 22u
a.
5.
du dx
What is the derivative with respect to x of sec x2? (M95 M 10) a. 2x sec x2 tan x2 c. sec x2 tan x2 b. 2x sec x tan x d. 2 sec x2 tan x2
sech x tanh x
Find the derivative of loga u with respect to x. (M94 M 19) a.
8.
√ 1 – x2 1
√ 1 – x2 Find y’ if y = ln x. (M94 M 24) a. 1/x b. ln x2
c. d.
1 1 + x2 1+x
d.
√ 1 – x2
c. d.
1 / ln x x ln x
c. d.
au / ln a a ln u
2
3 (x + 1)
+ x2 3 (x + 1)2 – x 3 (x + 1)2 x2
–
(x + 1)3 x2 (x + 1)3 x2 (x + 1)3 x2 (x + 1)3 x2
11.
Find the first derivative of y = arc cos 4x. (M97 M 15) a. -4 / (1 – 16x2)0.5 c. 4 / (1 – 4x2)0.5 b. 4 / (1 – 16x2)0.5 d. -4 / (1 – 4x2)0.5
12.
Find the first derivative of y = arc sin 3x. (N97 M 27) a. 3 / (1 – 3x2)0.5 c. 3 / (1 – 9x2)0.5 2 0.5 b. -3 / (1 – 9x ) d. -3 / (1 – 3x2)0.5
13.
What is the derivative of y = 33x? (N02 M 14) a. 3(3x + 1) ln 3 c. 3(3x – 3) ln 3 b. 33x ln 3 d. 33x ln (3x)
u
Find y’ if y = a . (M94 M 25) a. au ln a b. u ln a
3 (x + 1)2 + x
What is the derivative with respect to x of (x + 1)3 – x3? (N94 M 13) a. 3x + 3 c. 6x – 3 b. 3x – 3 d. 6x + 3
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MATHEMATICS DIFFERENTIAL CALCULUS 14.
What is the derivative with respect to x of 2 cos (2 + x3)? (M03 M 15) a. 6x2 sin (2 + x3) c. 2 sin (2 + x3) b. -6x2 sin (2 + x3) d. -2 sin (2 + x3)
SLOPE OF A CURVE 20.
Find the slope of the tangent to the curve y = 2x – x2 + x3 at x = 0. (M94 M 13) a. 2 c. 4 b. 3 d. 5
21.
Find the slope of the ellipse x2 + 4y2 – 10x – 16y + 5 = 0 at the point where y = 2 + 80.5 and x = 7. (M96 M 10) a. -0.1654 c. -0.1768 b. -0.1538 d. -0.1463
22.
What is the slope of the curve x2 + y2 – 6x + 10y + 5 = 0 at (1, 0)? (M98 M 24) 2 a. /5 c. -2/5 5 b. /2 d. -5/2
23.
Determine the slope of the curve x2 + y2 – 6x – 4y – 21 = 0 at (0, 7). (N98 M 11) 3 a. /5 c. -3/5 2 2 b. - /5 d. /5
24.
Determine the slope of the curve y = 6(4 + x)1/2 at point (0, 12). (N99 M 2) a. 3/2 c. 1/2 b. -3/2 d. -1/2
25.
At what value of x will the slope of the curve x3 – 9x – y = 0 be 18? (M00 M 13) a. 2 c. 5 b. 4 d. 3
HIGHER DERIVATIVES 15.
-2
Find the second derivative of y = x at x = 2. (M99 M 6) a. 96 c. -0.25 b. 0.375 d. -0.875
LIMITS OF FUNCTIONS 16.
17.
18.
Find the limit of the function (x2 – 1)/(x2 + 3x – 4) as x approaches 1. (N97 M 3) 1 a. 0 c. /4 2 2 b. /5 d. /3 Evaluate the limit of (x2 – 4)/(x – 2) as x approaches 2. (M01 M 24) a. 1 c. 3 b. 2 d. 4 Evaluate (M02 M 1) lim x 0 a. b.
19.
[
0 undefined
tan 2x – 2 sin x x3
]
c. d.
3 infinity
Evaluate the limit of (x – 4)/(x2 – x – 12) as x approaches 4. (N03 M 15) 1 a. /7 c. zero 1 b. undefined d. /6
TANGENT AND NORMAL LINES 26.
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What is the equation of the normal to the curve x2 + y2 = 25 at (4, 3)? (M95 M 6) a. 3x – 4y = 0 c. 5x – 3y = 0 b. 5x + 3y = 0 d. 3x + 4y = 0
MATHEMATICS DIFFERENTIAL CALCULUS 27.
What is the equation of the tangent to the curve 9x2 + 25y2 – 225 = 0 at (0, 3)? (M96 M 13) a. y+3=0 c. x–3=0 b. x+3=0 d. y–3=0
28.
Find the equation of a line normal to the curve x2 = 16y at (4, 1). (N96 M 14) a. 2x – y – 9 = 0 c. 2x – y + 9 = 0 b. 2x + y – 9 = 0 d. 2x + y + 9 = 0
29.
What is the equation of the tangent to the curve x2 + y2 = 41 at (5, 4)? (M97 M 29) a. 5x + 4y – 41 = 0 c. 5x – 4y – 41 = 0 b. 5x + 4y + 41 = 0 d. 5x – 4y + 41 = 0
EQUATION OF DIAMETER OF CURVES 30.
The chords of the ellipse 64x2 + 25y2 = 1600 having equal slopes of 1/5 are bisected by its diameter. Determine the equation of the diameter of the ellipse. (N98 M 28) a. 5x – 64y = 0 c. 5x + 64y = 0 b. 64x – 5y = 0 d. 64x + 5y = 0
31.
The chords of the ellipse 4x2 + 9y2 = 144 having equal slopes of 3/4 is bisected by its diameter. What is the equation of the diameter? (M01 M 17) a. 18x – 25y = 0 c. 24x + 17y = 0 b. 16x + 27y = 0 d. 14x – 31y = 0
VELOCITY AND ACCELERATION 32.
33.
The motion of a body moving vertically upwards is expressed as h = 100t – 16.1t2, where h is the height in feet and t is the time in seconds. What is the velocity of the body when t = 2 seconds? (N00 M 24) a. 21.7 fps c. 24.1 fps b. 28.7 fps d. 35.6 fps
TIME-RATES 34.
Gas is escaping from a spherical balloon at a constant rate of 2 ft3/min. How fast, in ft2/min, is the outer surface area of the balloon shrinking when the radius is 12 ft? (N94 M 15) a. 2 c. 0.333 b. 3 d. 0.5
35.
Two railroad tracks are perpendicular to each other. At 12:00 p.m. there is a train at each track approaching the crossing at 50 kph, one being 100 km, the other 150 km away from the crossing. How fast in kph is the distance between the two trains changing at 4:00 p.m.? (M96 M 8) a. 68.08 c. 69.08 b. 67.08 d. 70.08
36.
A car starting at 12:00 noon travels west at a speed of 30 kph. Another car starting from the same point at 2:00 p.m. travels north at 45 kph. Find how fast the two are separating at 4:00 p.m. (N96 M 27) a. 55 c. 57 b. 51 d. 53
37.
A car drives east from point A at 30 kph. Another car 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? (M97 M 16) a. 74 c. 45 b. 78 d. 54
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/min will it be moving at the end of 4 minutes? (M98 M 14) a. 39.8 c. 29.8 b. 49.8 d. 19.8 ------- 3 -------
MATHEMATICS DIFFERENTIAL CALCULUS 38.
A lighthouse is 2 km off a straight shore. A searchlight at the lighthouse focuses to a car moving along the shore. When the car is 1 km from the point nearest to the lighthouse, the searchlight rotates 0.25 rev/hour. Find the speed of the car in kph. (M00 M 17) a. 3.93 c. 2.92 b. 2.56 d. 3.87
39.
A 3-m 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? (N02 M 16) a. 1.81 c. 1.79 b. 1.66 d. 1.98
43.
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. (N01 M 8) a. 0.02 c. 0.08 b. 0.05 d. 0.12
44.
Water flows at the rate of 16 m3/min in a conical tank 12 m diameter on top and 24 m deep. When the water in the tank is h meters deep, the surface is rising at the rate of 0.566 m/min. Find the value of h. (N01 M 30) a. 10 m c. 12 m b. 14 m d. 8m
FLOW RATES
MAXIMA AND MINIMA
40.
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? (M95 M 12) a. 3 c. 2 b. 5 d. 4
45.
41.
There is a constant inflow of a liquid into a conical vessel 15 ft deep and 7.5 ft in diameter at the top. Water is rising at the rate of 2 ft/min when the water is 4 ft deep. What is the rate of inflow in ft3/min? (N98 M 9) a. 8.14 c. 9.33 b. 7.46 d. 6.28
A rectangular corral is to be built with a required area A. If an existing fence is to be used as one of the sides, determine the relation of the width and the length which would cost the least. (M94 M 18) a. width = twice the length b. width = 1/2 length c. width = length d. width is three times the length
46.
A cylindrical steam boiler is to have a volume of 1340 ft3. The cost of the metal sheets to make the boiler should be a minimum. What should be its base diameter in feet? (N94 M 14) a. 7.08 c. 11.95 b. 8.08 d. 10.95
47.
A wall h meters high is 2 m away from a 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? (M95 M 11) a. 2.24 c. 2.14 b. 2.44 d. 2.34
42.
Water flows at the rate of 16 m3/min in a conical tank 12 m diameter on top and 24 m deep. How fast is the water surface rising when the water is 12 m deep in the tank? (M01 M 4) a. 0.231 m/min c. 0.828 m/min b. 0.712 m/min d. 0.566 m/min
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MATHEMATICS DIFFERENTIAL CALCULUS 48.
A manufacturer can produce a commodity at a cost of P2.00 per unit. At a selling price of P5.00 each consumer have been buying 4,000 pieces a month for each price. P1.00 increase in the price, 400 fewer pieces will be sold each month. At what price a piece will the total profit be maximized? (N95 M 10) a. P9.50 c. P7.75 b. P8.50 d. P9.50
49.
The speed of the traffic flowing past a certain downtown exit between the hours of 1:00 p.m. and 6:00 p.m. 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 p.m. and 6:00 p.m. in mph? (M96 M 9) a. 50 c. 48 b. 46 d. 52
50.
51.
52.
53.
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. (M99 M 7) a. 9850 c. 10200 b. 9800 d. 7500
54.
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. (M99 M 8) a. 16.14 cm c. 18.41 cm b. 32.28 cm d. 28.74 cm
55.
The volume of a closed cylindrical tank is 11.3 m3. If the total surface area is a minimum, what is its base radius, in meters? (N99 M 9) a. 1.44 c. 1.22 b. 1.88 d. 1.66
56.
What is the least amount of tin sheet, in square inches, that can be made into a closed cylindrical can having a volume of 108 in3? (M97 M 22) a. 125.5 c. 127.5 b. 123.5 d. 129.5
The total surface area of a closed cylindrical tank is 153.94 m2. If the volume is to be maximum, what is its height in meters? (M00 M 18) a. 6.8 m c. 3.6 m b. 5.7 m d. 4.5 m
57.
The volume of a closed cylindrical tank is 11.3 m3. If the total surface area is a minimum, what is its base radius, in meters? (M98 M 19) a. 1.44 c. 1.22 b. 1.88 d. 1.66
A closed cylindrical tank having a volume of 71.57 m3 is to be constructed. If the surface area is to be a minimum, what is the required diameter of the tank? (N00 M 11) a. 4m c. 5m b. 5.5 m d. 4.5 m
58.
The sum of two numbers is S. What is the minimum sum of their cubes? (N01 M 29) a. S3 / 3 c. S3 / 2 3 b. S /4 d. S3 / 5
59.
Determine the shortest distance from point (4, 2) to the parabola y2 = 8x. (M02 M 28) a. 2.83 c. 2.41 b. 3.54 d. 6.32
A Norman window has 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? (N96 M 19) 2 a. 1 c. /3 1 1 b. /3 d. /2
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MATHEMATICS DIFFERENTIAL CALCULUS 60.
Suppose that x years after founding in 1975, a certain employee association had a membership of f(x) = 100(2x3 – 45x2 + 264x). At what time between 1975 and 1989 was the membership smallest? (N02 M 15) a. 1983 c. 1984 b. 1985 d. 1986
61.
The sum of two numbers is C. The product of one by the cube of the other is to be a maximum. Determine one of the numbers. (N03 M 16) a. 3C/4 c. 3C/2 b. 3C/8 d. 3C/7
62.
A triangular lot ABC has AB = 4.25 m, BC = 9.61 m, and CA = 8.62 m. A rectangular lot is inscribed in it such that the shorter side is on the 4.25 m side of the triangle. Determine the maximum area of the rectangular lot. (N03 M 17) a. 12.32 m2 c. 8.24 m2 2 b. 9.16 m d. 7.12 m2
66.
A particle moves according to the parametric equations: y = 2t2 x = t3 where x and y are displacements (in meters) in x and y direction, respectively, and t is time in seconds. Determine the acceleration of the body after t = 3 seconds. (M02 M 11) a. 12.85 m/s2 c. 21.47 m/s2 2 b. 18.44 m/s d. 5.21 m/s2
67.
A particle moves according to the parametric equations x = t3 and y = 2t2. What is the velocity of the particle when t = 2. (N03 M 18) a. 15.12 c. 14.42 b. 13.21 d. 16.89
CURVATURE 68.
What is the curvature of the curve y2 = 16x at the point (4, 8)? (M03 M 16) a. -0.044 c. -0.066 b. -0.088 d. -0.033
PARAMETRIC EQUATIONS 63.
64.
65.
Find the slope of the line whose parametric equations are x = 2 + t and y = 5 – 3t. (M97 M 27) a. -3 c. 2 b. 3 d. -2 Find the slope of the line whose parametric equations are x = -1 + t and y = 2t. (N97 M 24) a. -2 c. -1 b. 1 d. 2 Find the slope of the line whose parametric equations are x = 4t + 6 and y = t – 1. (M98 M 8) a. -4 c. 4 1 b. /4 d. -1/4
RADIUS OF CURVATURE 69.
Determine the radius of curvature at (4, 4) of the curve y2 – 4x = 0. (N97 M 7) a. 23.4 c. 25.4 b. 22.4 d. 24.4
70.
What is the radius of curvature at point (1, 2) of the curve 4x – y2 = 0? (N98 M 26) a. 6.21 c. 5.66 b. 5.21 d. 6.66
71.
Find the radius of curvature of the curve x = y3 at (1, 1). (N99 M 16) a. 4.72 c. 4.67 b. 3.28 d. 5.27
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