diffcal

1. 1. Anelliptical plotof gardenhasa semi-majoraxisof 6mand a semi-minoraxisof 4.8meters.If these
are increasedby0.15m each, findbydifferential equationsthe increase inareaof the gardenin sq.m.
a. 0.62π b. 1.62π c. 2.62π d. 2.62π
2. The altitude of aright circularcylinderistwice the radiusof the base.The altitude ismeasuredas
12cm. Witha possible errorof 0.005cm, findthe approximatelyerrorinthe calculatedvolume of the
cylinder.
a. 0.188 cu cm b. 0.144 cu cm c. 0.104 cu cm d. 0.126 cu cm
3. If y=x^(3/2) whatis the approximate change inywhenx changesfrom9 to 9.01?
a. 0.045 b. 0.068 c. 0.070 d. 0.023
4. If y=3x^2-x+1, findthe pointx at whichdy/dx assume itsmeanvalue inthe intervalx=2and x=4.
a. 3 b. 6 c. 4 d. 8
5. Findthe derivativeof y=2^(4x).
a. 3^(4x+2) ln2
b. 2^(4x+2) ln 2
c. 6^(3x+2) ln2
d. 4^(4x+2) ln 2
6. Findthe derivativeof arccos (2x).
a. -2/√(1-4x^2)
b. 2/√(1-4x^2)
c. 2/(1+4x^2)
d. 2/√(2x^2-1)
7. Findthe derivativeof 4 arc tan (2x).
a. 4/(1+x^2)
b. 4/(4x^2+1)
c. 8/(1+4x^2)
d. 8/(4x^2+1)
8. Findthe derivative of arccsc (3x).
a. -1/[x√(9x^2-1)]
b. 1/[3x√(9x^2-1)]
c. 3/[x√(1-9x^2)]

2. d. 3/[x√9x^2-1)
9. Determine the slope of the curve y=x^2-3x as itpassesthrough the origin.
a. -4 b. 2 c. -3 d. 0
10. Findthe slope of (x^2)y=8 at the point(2,2).
a. 2 b. -1 c. -1/2 d. -2
11. Finddy/dx byimplicitdifferentiationatthe point(3,4) whenx^2+y^2=25.
a. -3/4 b. ¾ c. 2/3 d. -2/3
12. Findthe pointof inflectionof f(x)=x^3-3x^2-x+7.
a. 1,4 b. 1,2 c. 2,1 d. 3,1
13. Findthe pointof inflectionof the curve y=(9x^2-x^3+6)/6.
a. 3,10 b. 2,8 c. 3,8 d. 2,10
14. Findthe pointof inflectionof the curve y=x^3-3x^2+6.
a. 1,4 b. 1,3 c. 0,2 d. 2,1
15. Compute the radiusof curvature of the parabolax^2=4y at the point(4,4).
a. 22.36 b. 24.94 c. 20.38 d. 18.42
16. Findthe radiusof curvature of the curve y=2x^3+3x^2 at (1,5).
a. 97 b. 90 c. 101 d. 87
17. A rectangularcorral is to be builtwitha requiredarea.If anexistingfence istobe usedasone of the
sides,determine the relationof the widthandthe lengthwhichwouldcostthe least.
a. width=twice the length
b. width=1/2length
c. width=length
d. width=3timesthe length
18. Findthe twonumberswhose sumis20, if the product of one by the cube of the otheris to be
minimum.
a. 5 and 15 b. 10 and 10 c. 4 and 16 d. 8 and 12
19. The sum of two numbersis12. Findthe minimumvalue of the sumof their cubes.
a. 432 b. 644 c. 346 d. 244

3. 20. A printedpage mustcontain60 sqm of printedmaterial.There are tobe marginsof 5cm on either
side andmarginsof 3cm on top and bottom.How longshouldthe printedlinesbe inordertominimize
the amountof paperused?
a. 10 b. 18 c. 12 d. 15
21. A rectangularbox withsquare base andopenat the top is tohave a capacityof 16823 cu cm. Find
the heightof the box that requiresminimumamountof materialsrequired.
a. 16.14 cm b. 14.12 cm c. 12.13 cm d. 10.36 cm
22. A closedcylindrical tankhasa capacityof 576.56 cu m. Findthe minimumsurface areaof the tank.
a. 383.40 cu m b. 412.60 cu m c. 516.32 cu m d. 218.60 cu m
23. A wall 2.245 m highis x metersawayfroma building.The shortestladderthatcanreach the building
withone endrestingonthe groundoutside the wall is6m. What isthe value of x?
a. 2m b. 2.6m c. 3.0m d. 4.0m
24. Withonly381.7 sq m of materials,aclosedcylindrical tankof maximumvolume istobe the heightof
the tank, inm?
a. 9m b. 7m c. 11m d. 13m
25. If the hypotenuseof aright triangle isknown,whatisthe ratioof the base and the altitude of the
righttriangle whenitsare is maximum?
a. 1:1 b. 1:2 c. 1:3 d. 1:4
26. What isthe maximumlengthof the perimeterif the hypotenuse of arighttriangle is5m long?
a. 12.08 m b. 15.09 m c. 20.09 m d. 8.99 m
27. An opentoprectangulartank withsquare sbasesis to have a volume of 10 cu m.The material firits
bottomisto cost 15 centspersq m and that forthe sides6cents persq m. Findthe mosteconomical
dimensionsforthe tank.
a. 2 x 2 x 2.5 b. 2 x 5 x 2.5 c. 2 x 3 x 2.5 d. 2 x 4 x 2.5
28. Divide the number60 intotwopats so that the product Pof one part and the square of the otheris
maximum.Findthe smallestpart.
a. 20 b. 22 c. 10 d. 27
29. The edgesof a rectangularbox are to be reinforcedwithanarrow metal strips.If the box will have a
volume of 8 cu m, whatwoulditsdimensionsbe torequire the leasttotal lengthof strips?
a. 2 x 2 x 2 b. 4 x 4 x 4 c. 3 x 3 x 3 d. 2 x 2 x 4
30. A rectangularwindowsurmountedbyaright isoscelestrianglehasa perimeterequal to54.14m.
Findthe heightof the rectangularwindow sothatthe window willadmitthe mostlight.
a. 10 b. 22 c. 12 d. 27

4. 31. A normal windowisinthe shape of a rectangle surroundedbyasemi-circle.If the perimeterof the
windowis71.416, what isitsradiusand the heightof the rectangularportionsothat it will yielda
windowadmittingthe mostlight?
a. 10 b. 22 c. 12 d. 27
32. Findthe radiusof a right circularcone havinga lateral areaof 544.12 sq m to have a maximum
volume.
a. 10 b. 20 c. 17 d. 19
33. A rectangularlothas an area of 1600 sq m. Findthe leastamountof fence thatcould be usedto
enclose the area.
a. 160m b. 200m c. 100m d. 300m145.
34. A wall 2.245m highis2m awayfrom a bldg.Findthe shortestladderthatcan reach the bldgwithone
endrestingonthe ground outside the wall.
a. 6m b. 9m c. 10m d. 4m
35. The hypotenuseof aright triangle is20cm. What isthe max possible areaof the triangle insqcm?
a. 100 b. 170 c. 120 d. 160
36. A rectangularfieldhasanarea of 10,000 sqm. What is the leastamountof fencingmetersto
enclose it?
a. 400 b. 370 c. 220 d. 560
37. Findthe radiusof the circle inscribe ina triangle havingamax area of 173.205 sq cm.
a. 3.45 cm b. 5.77 cm c. 4.96 cm d. 2.19 cm
38. The areaof a circle inscribe ina triangle isequal to113.10 sq cm. Findthe max area of the triangle.
a. 186.98 sq cm b. 156. 59 sq cm c. 175.80 sq cm d. 193. 49 sq cm
39. Water isthe flowingintoaconical vessel 15cmdeepandhavinga radiusof 3.75cm across the top.If
the rate at whichwateris risingis2cm/s,how fastisthe waterflowingintothe conical vessel whenthe
depthof wateris 4cm?
a. 6.28 cu m/min b. 4 cu m/min c. 2.5 cu m/min d. 1.5 cu m/min
40. Two sidesof a triangle are 5 and8 unitsrespectively.If the includedangleischangingatthe rate of
one radianpr second,at what rate is the thirdside changingwhenthe includedangleis60 degrees?
a. 4.95 units/sec b. 5.55 units/sec c. 4.24 units/sec d. 3.87 units/sec
41. The two adjacentsidesof a triangle are 5 and 8 metersrespectively.If the includedangleischanging
at the rate of 2 rad/sec,at what rate isthe area of the triangle changingif the includedangleis60
degrees?
a. 20 sq m/sec b. 25 sq m/sec c. 15 sq m/sec d. 23 sq m/sec

5. 42. A triangulartroughis12m long,2m wide at the top and2m deep.If waterflowsinatthe rate of 12
cu m per min,findhowfastthe surface isrisingwhenthe wateris1m deep.
a. 1 b. 2 c. 3 d. 4
43. Water flowsintoavertical cylindrical tank,atthe rate of 1/5 cu ft/sec.The watersurface is risingat
the rate of 0.425ft/min. What isthe diameterof the tank?
a. 6 ft b. 10 ft c. 8 ft d. 4 ft
44. The radiusof a sphere ischangingata rate of 2 cm/sec.Findthe rate of change of the surface area
whenthe radiusis6cm.
a. 96π sq cm/sec b. 78π sq cm/sec c. 84π sqcm/sec d. 68π sq cm/sec
45. A spherical balloonisinflatedwithgasat the rate of 20 cu m/min.How fast isthe radiusof the
balloonchangingatthe instantthe radiusis 2cm?
a. 0.398 b. 0.422 c. 0.388 d. 0.498
46. The base radiusof a cone ischangingat a rate of 3cm/sec.Findthe rate of change of its volume
whenthe radiusis4cm and itsaltitude is6cm.
a. 48 π cu cm/sec b. 24 π cu cm/sec c. 18 π cu cm/sec d. 36 π cu cm/sec
47. Water isflowingintoafrustumof a cone at a rate of 100 liter/min.The upperradiusof the frustum
of a cone is 1.5m while the lowerradiusis1mand a heightof 2m. If the waterrises at the rate of
0.04916 cm/sec,findthe depthof water.
a. 15.5cm b. 10.3cm c. 13.6cm d. 18.9cm
48. Water isflowingintoaconical vessel 18cm deepand10 cm across the top. If the rate at whichthe
watersurface isrisingis27.52 mm/sec,how fast is the waterflowingintothe conical vessel whenthe
depthof wateris 12cm?
a. 9.6 cu cm/sec b. 7.4 cu cm/sec c. 8.5 cu cm/sec d. 6.3 cu cm/sec
49. a statue 3m highisstandingona base of 4m high.if anobserver'seye is1.5m above the ground,how
far shouldhe standfromthe base inorderthat the angle subtendedbythe statue isa minimum?
50. AnLRT train6 m above the groundcrossesa streetat a speedof 9 m/s,at the instantthata car
approachingat a speedof 4 m/sis12 m upthe street.Findthe rate of the LRT train andthe car are
separatingone secondlater.
a. 3.64 m/s b. 4.34 m/s c. 6.43 m/s d. 4.63 m/s
51. Car A movesdue eastat 30kph at the same instantcar B ismovingS30deg E.witha speedof 30kph.
The distance fromA to B is 30km. Findhow fast isthe speedbetweenthemare separatingafterone
hour.
a. 45 kph b. 36 kph c. 40 kph d. 38 kph