MATHEMATICS
DIFFERENTIAL CALCULUS
DIAGNOSTIC EXAMINATION - QUESTIONNAIRE
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DIFFERENTIAL CALCULUS
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1. THE VELOCITY OF AN AUTOMOBILE STATING FROM REST IS GIVEN BY ds/dt= 90t/(t+10)
ft/sec, DETERMINE ITS ACCELERATION AFTER AN INTERVAL OF 10 sec. (in ft/sec2)
A.
B.
C.
D.
2.25
3.25
2.52
3.52
2. AN IRON BAR 10m LONG IS BENT TO FORM A CLOSED PANE AREA. WHAT IS THE
LARGEST AREA POSSIBLE
A. 32.83
B. 30.83
C. 31.83
D. 33.83
3. A MANUFACTURER ESTIMATES THAT THE COST OF THE PRODUCTION OF X UNITS OF A
CERTAIN ITEM IS C=40X-0.02X^2-600. HOW MANY UNITS SHOULD BE PRODUCED FOR
MINIMUM COST?
A.
B.
C.
D.
100
1000
10000
100000
4. GIVEN THE COST EQUATION OF A CERTAIN PRODUCT AS FOLLOWS C= 50T^2-200T+10000
WHERE T IS IN YEARS. FIND THE MAXIMUM COST FROM THE YEAR 1995 TO 2002.
A.
B.
C.
D.
9800
8800
7800
10800
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5.
FIND THE RADIUS OF A RIGHT CIRCULAR CONE HAVING A LATERAL AREA OF 544.12
SQ. M, TO HAVE A MAXIMUM VALUE.
A.
B.
C.
D.
10
11
12
13
6. A RECTANGULAR LOT HAS AN AREA OF 1600 SQ. M, FIND THE LEAST AMOUNT OF FENCE
THAT COULD BE USED TO ECNLOSE THE AREA.
A.
B.
C.
D.
140 m
150 m
160 m
170 m
7.
A VEHICLE MOVES ALONG TRAJECTORY HAVING COORDINATED GIVEN AS X = T^3 AND
Y= 1 –T^2. THE ACCELERATION OF THE VEHICLE AT ANY PONT OF THE TRAJECTORY IS A
VECTOR HAVING MAGNITUDE AND DIRECTION. FIND THE ACCELERATION WHEN T =2.
A.
B.
C.
D.
12
12.17
13
13.17
8. A PARTICLE MOVES ALONG A PATH WHOSE PARAMETRIC EQUATIONS ARE X= T^3 AND
Y = 2T^2. WHAT IS THE ACCELERATION OF THAT PARTICLE WHEN T=5 SECONDS?
A. 30.26 m/s2
B. 18.56 m/s2
C. 23.37 m/s2
D. 21.62 m/s2
9.
FIND THE MINIMUM AMOUNT OF THIN SHEET THAT CAN BE MADE INTO A CLOSED
CYLINDER HAVING A VOLUME OF 108 CU IN. IN SQUARE INCHES.
A. 125.5
B. 129.5
C. 127.5
D. 123.5
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10.
A TIME STUDY SHOWD THAT ON AVERAGE, THE PRODUCTIVITY OF A WORKER
AFTER T HOURS ON THE JOB CAN BE MODELED BY THE EXPRESSION P= 27+6T-T^3
WHERE P IS THE NUMBER OF UNITS PRODUCED PER HOUR. WHAT IS THE MAXIMUM
PRODUCTIVITY EXPECTED?
A. 44
B. 34
C. 33
D. 40
11.
THE TOTAL COST OF PRODUCTION SPARE PARTS OF COMPUTERS IS GIVEN AS C
=4000X-100X^2+X^3 WHERE X IS THE NUMBER OF UNITS OF SPARE PARTS PRODUCED SO
THAT THE AVERAGE COST WILL BE MINIMUM?
A. 50
B. 20
C. 4
D. 20
12.
IF THE HYPOTENUSE OF A RIGHT TRIANGLE IS KNOWN, WHAT IS THE RATIO OF THE
BASE AND THE ALTITUDE OF THE RIGHT TRIANGLE WHEN ITS AREA IS MAXIMUM?
A. 1:1
B. 1:4
C. 1:3
D. 1:2
13.
A PARTICLE MOVES ALONG A PATH WHOSE PARAMETRIC EQUATIONS ARE X= T^3
AND Y = 2T^2. WHAT IS THE ACCELERATION OF THAT PARTICLE WHEN T=3 SECONDS?
A. 30.26 m/s2
B. 18.44 m/s2
C. 23.37 m/s2
D. 21.62 m/s2
14. THE COST PER UNIT OF PRODUCTION IS EXPRESSED AS (4+3X) AND THE SELLING
PRICE ON THE MARKET IS P100 PER UNIT. WHAT IS THE MAXIMUM DAILY PROFIT THAT
THE COMPANY SAN EXPECT OF THIS PRODUCT?
A.
B.
C.
D.
P 876
P 768
P 657
P 678
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15.
THE TOTAL COST OF PRODUCTION A SHIPMENT OF A CERTAIN PRODUCT IS C =
5000X+125000/X WHERE X IS THE NUMBER OF MACHINES USED IN THE PRODUCTION. HOW
MANY MACHINES WILL MINIMIZE THE TOTAL COST?
A.
B.
C.
D.
15
5
10
20
16.
THE DEMAND X FOR A PRODUCT IS X = 10000-100P WHERE P IS THE MARKET PRICE
IN PESOS PER UNIT. THE EXPENDITURE FOR THE TWO PRODUCT IS E =PX. WHAT MARKET
PRICE WILL THE EXPECNDITURE BE THE GREATEST?
A. 60
B. 70
C. 100
D. 50
17. FIND THE DERIVATIVE OF SIN(3X^2-X)
A. 𝑠𝑖𝑛(3𝑥 2 − 𝑥)cos⁡(6𝑥 − 1)
B. cos⁡(6𝑥 − 1)
C. (6𝑥 − 1) cos(3𝑥 2 − 𝑥)
D. (6𝑥 − 1) sin(3𝑥 2 − 𝑥)
18. FIND THE DERIVATIVE OF 𝒔𝒊𝒏(𝒆𝒙 )
A. cos(𝑥 𝑥−1 )
B. sin(𝑥𝑒 𝑥−1 )
C. 𝑒 𝑥 cos⁡(𝑒 𝑥 )
D. cos⁡(𝑒 𝑥 )
19. FIND 𝐥𝐢𝐦
𝒙𝟐 −𝟐𝟓
𝒙→𝟓 𝒙−𝟓
A. 5
B. 10
C. 15
D. 0
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Find 𝐥𝐢𝐦
20.
𝒙→𝟎 𝒙+𝟏
A.
B.
C.
D.
21.
𝟓𝒙𝟐 −𝟒
-4
4
-3.99
3.99
𝟑𝒙𝟐 +𝟏
Find 𝐥𝐢𝐦 𝟐𝒙𝟐 −𝒙
𝒙→∞
A.
B.
C.
D.
2/3
3/2
-2/3
-3/2
22. WHAT IS THE LIMIT OF
𝟒𝒙𝟑 +𝒙
𝒙𝟐 −𝟐𝒙𝟑
AS X APPROACHED INFINITY?
A. -2
B. 2
C. 4
D. -4
23.
FIND 𝐥𝐢𝐦
−𝟒𝒙𝟑 −𝟏
𝟑𝒙𝟐
𝒙→∞
A. −∞
B. ∞
C. 0
D. DNE
24.
FIND 𝐥𝐢𝐦
𝒙→+∞
𝟐𝒙+𝟓
𝒙𝟐 −𝟕𝒙+𝟑
A. −∞
B. ∞
C. 0
D. DNE
25.
FIND 𝐥𝐢𝐦
𝒙𝟔 −𝟑𝒔𝒊𝒏𝒙
𝒙→∞ 𝟑𝒙𝟐 +𝟒𝒙𝟔
A. −∞
B. −∞
C. 0
D. 1/4
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26. FIND 𝐥𝐢𝐦
𝟏−𝒄𝒐𝒔𝒙
𝒙→𝟎
𝒙𝟐
APPLY L’HOSPITAL’S RULE
A. 1/3
B. 1/2
C. 1/4
D. 1
27.
𝒙𝟑
BY APPLYING L’HOSPITAL’S RULE, FIND 𝐥𝐢𝐦 𝒆𝒙
𝒙→∞
A.
B.
C.
D.
28.
1
2
0
5
Differentiate: 𝑦=3^4𝑥
A. 81
B. 81^x
C. ln⁡(81)
D. 81𝑥 ln⁡(81)
29.
If (𝑥)=〖5−12𝑥−2𝑥〗^2, prove that f satisfies the hypotheses of the Rolle’s Theorem on the
interval [-7,1] and find the number 𝑐∈(−7,1) that satisfies the condition of the theorem.
A. 120
B. 125
C. 150
D. 175
30.
WHAT IS THE SLOPE OF THE CURVE 𝒙𝟐 + 𝒚𝟐 − 𝟔𝒙 + 𝟏𝟎𝒚 + 𝟓 = 𝟎 AT (1,0)?
A. 2/3
B. 2/5
C. 3/2
D. 4/3
31.
GIVEN THE FUNCTION F(X)= X TO THE 3RD POWER – 6X + 2, FIND THE VALUE OF THE
FORST DERIVATIVE AT X=2. F’(2).
A. 6
B. 7
C. 3X^2-5
D. 8
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32.
FIND THE DERIVATIVES WITH RESPECT TO X OF THE FUNCTION √(𝟐 − 𝟑𝑿𝟐 )
A. −3𝑋/√(2 − 3𝑋 2 )
B. −2𝑋 2 /√(2 + 3𝑋 2 )
C. −3𝑋/√(2 + 3𝑋 2 )
D. −2𝑋 2 /√(2 − 3𝑋 2 )
33. FIND THE DERIVATIVE OF (X+5)/(X^2-1) WITH RESPECT TO X
A. 𝐷𝐹 (𝑥) = (𝑥 2 + 10𝑥 − 1)/(𝑥 2 − 1)2
B. 𝐷𝐹(𝑥) = (−𝑥 2 − 10𝑥 + 1)/(𝑥 2 − 1)2
C. 𝐷𝐹(𝑥) = (𝑥 2 − 10𝑥 − 1)/(𝑥 2 − 1)2
D. 𝐷𝐹(𝑥) = (−𝑥 2 − 10𝑥 − 1)/(𝑥 2 − 1)2
34.
FIND THE SLOPE OF THE CURVE Y=6(4+X)^1/2 AT POINT (0,12).
A. 2.8
B. 2.2
C. 1.8
D. 1.5
35.
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 20INCHES AND
INCREASES AT THE RATE OF ONE INCH PER SECOND . AT WHAT RATE IS THE VOLUME
CHANGING?
A. 1257 in3/sec
B. 1347 in3/sec
C. 1812 in3/sec
D. 1557 in3/seC
36.
FIND THE APPROXIMATE CHANGE IN VOLUME OF A CUBE OF SIDE X CAUSED BY
INCREASING THE SIDES BY 2%.
A. 5%
B. 6%
C. 7%
D. 8%
37. WATER IS FLOWING INTO A CONICAL RESERVOIR 20FT DEEP AND 10FT ACROSS THE
TOP AT A RATE OF 15〖𝑓𝑡〗^3/𝑚𝑖𝑛. FIND HOW FAST IS THE SURFACE RISING WHEN THE
WATER IS 8FT DEEP.
𝟏𝟓
A. 𝟒𝝅 𝒇𝒕/𝒎𝒊𝒏
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𝟏
B. 𝟐𝝅 𝒇𝒕/𝒎𝒊𝒏
𝟓
C. 𝟒𝝅 𝒇𝒕/𝒎𝒊𝒏
𝟏𝟑
D.⁡𝟒𝝅 𝒇𝒕/𝒎𝒊𝒏
38. A LADDER 10FT LONG LEANS AGAINST A WALL. FIND THE RATE OF THE ANGLE 𝜃
BETWEEN THE LADDER AND THE GROUND IS CHANGING AT THE GIVEN MOMENT WHEN THE
LADDER IS 6FT AWAY FROM THE WALL AND SLIDING ALONG THE GROUND AWAY FROM
THE WALL AT THE RATE OF 2FT/S.
𝟏
A. − 𝒓𝒂𝒅/𝒔
𝟑
B.
𝟏
𝟒
𝒓𝒂𝒅/𝒔
𝟏
C. − 𝟒 𝒓𝒂𝒅/𝒔
𝟏
D. 𝟑 𝒓𝒂𝒅/𝒔
39.
A LADDER 20 FT LONG LEANS AGAINST A VERTICAL WALL. IF THE TOP SLIDES
DOWNWARD AT A RATE OF 2FT/S, FIND HOW FAST THE LOWER END IS MOVING WHEN IT IS
16FT FROM THE WALL.
A. 12
B. 1.4
C. 1.2
D. 1.5
40. FIND THE COORDINATES OF THE VERTEX OF THE PARABOLA 𝑦= 𝑥^2−4𝑥+1 BY MAKING
USE OF THE FACT THAT AT THE VERTEX, THE SLOPE OF THE TANGENT IS ZERO.
A. (3, 2)
B. (2, -3)
C. (-3, -2)
D. (-2, 3)
41. FIND THE EQUATION OF THE TANGENT TO THE CURVE 𝑦=𝑥+2𝑥^(1/3) THROUGH POINT
(8,12).
A. 7x – 6y + 16 =0
B. 6x -7y +18 =0
C. 6y – 7x +16 = 0
D. -7x -6y -16 =0
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43.
WHAT IS THE EQUATION NORMAL TO THE CURVE X^2 + Y^2 =25 AT (4,3).
A. 4x -3y =0
B. -3x – 4y =0
C. 3x + 4y =0
D. 3x – 4y =0
44.
THE SUM OF TWO POSITIVE NUMBERS IS 50. WHAT ARE THE NUMBERS IF THEIR
PRODUCT IS TO BE THE LARGEST POSSIBLE?
A. 20
B. 25
C. 30
D. 35
IF 𝑥=𝑒^𝑡 AND 𝑦=𝑒^𝑡 𝑠𝑖𝑛𝑡, FIND (𝑑^2 𝑦)/〖𝑑𝑥〗^2
45.
A.
B.
C.
D.
46.
−𝑠𝑖𝑛𝑡+𝑐𝑜𝑠𝑡
𝑒𝑡
𝑠𝑖𝑛𝑡+𝑐𝑜𝑠𝑡
𝑒𝑡
−𝑠𝑖𝑛𝑡−𝑐𝑜𝑠𝑡
𝑒𝑡
−𝑠𝑖𝑛𝑡+𝑐𝑜𝑠𝑡
−𝑒 𝑡
EVALUATE: lim (tan33x)/x3 AS X APPROACHES ZERO.
A. 27
B. 0
C. Infinity
D. 31
47.
EVALUATE 𝐥𝐢𝐦
𝒕𝒂𝒏𝒙
𝒙→𝟎
𝒙
A. 1
B. 0
C. UNDEFINED
D. INFINITY
48. If (𝑥,𝑦)=(𝑥𝑦^2)/(𝑥^2+𝑦^2 ), find 𝑓_𝑥𝑦 at (1,1)
A. 1/5
B. 1/2
C. 2/3
D. 5
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49.
FIND THE RADIUS OF CURVATURE AT ANY POINT OF THE CURVE Y + LN (COS X) = 0
A. sinx
B. cosx
C. cscx
D. secx
50.
FIND THE RADIUS OF CURVATURE OF THE FUNCTION 𝑦^2 = 8X AT (2,4)
A. 9.2
B. 9.3
C. 11.31
D. 13.11
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