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Direction: Encircle the letter of corresponding to your answer in the answer sheet provided.

1. At the minimum point, the slope of the tangent line to a curve is __________. a. positive b. negative c. zero d. infinity

2. If 1)( += xexf , then )1('f is equal to __________.

a. 0 b. 1 c. 1 d.

3. The volume of a cube is increasing at a rate of 6 cm3/min. How fast is the surface area increasing when the length of each edge is 12 cm? a. 2 cm2/min b. 4 cm2/min c. 6 cm2/min d. 8 cm2/min

4. Find the slope of the line tangent to x

y 4= at 2=x .

a. 2 b. 1 c. 1 d. 2

5. Evaluate: 352634lim 3

23

++

+ xx

xx

x

a. b. 0 c. d. 2

6. A rectangular field is fenced off, an existing wall being used as one side. If the area of the field is 7,200 ft2, find the least amount of fencing materials needed. a. 240 ft b. 250 ft c. 260 ft d. 270 ft

7. The rate of change of the area of a circle with respect to its radius when the diameter is 6 cm is _______________.

a. cm

cm2

4pi b. cm

cm2

5pi c. cm

cm2

6pi d. cm

cm2

7pi

8. If 233 =+ yxxy , find dxdy

at the point (1,1).

a. 2 b. 1 c. 1 d. 2

9. Find C so that the line 34 += xy is tangent to Cxy += 2 a. 7 b. 6 c. 5 d. 4

10. A man is walking at a rate of 1.5 m/s towards a street light which is 5 m above the level ground. At what rate is the tip of his shadow moving if the man is 2 m tall? a. 1.5 m/s b. 2.5 m/s c. 3.5 m/s d. 4.5 m/s

11. If ( )xexy 2ln= , find "y . a. 2

1x

b. 22x

c. x

1 d.

x

2

12. Find dxdy

for ( ) xyy lnlnlnln =+

a. yx

y+

b. yx

y

c. yx

x

+ d.

xyx

13. Two corridors are 6 m wide intersect at right angles. Find the length of the longest ladder that will go horizontally around the corner.

a. 26 m b. 212 m c. 218 m d. 224 m

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14. Consider the function: 3

2+

=x

y . The domain of the function is __________.

a. D = {x/x } b. D = {x/x 3} c. D = {x/x 3} d. D = {x/x 0}

15. A rectangular box with a square base is to have a capacity of 27 cu. in. Determine the least amount of material required a. 16 b. 54 c. 32 d. 72

16. Find the seventh derivative of the function: 6 5 4 3 27 2y x x x x x= + + a. 720 b. x 840 c. 0 d. 1

17. A statue 5 ft. fall stands on a pedestal 9 ft high. If an observers eye is 5 ft. above the ground, how far from the pedestal should he stand so that the angle subtended in his eye by the statue will be maximum. a. 36 b. 6 c. 9 d. 49

18. What positive number when raised to a power equal itself will give a minimum value?

a. e b. ln e c. 1e d. ln

1e

19. The depth of a conical vessel containing water is equal to the radius of its top. If the water leaks through a hole at the

vertex of the vessel at the rate of pi cu. ft/min, how fast is the surface of the water falling when the depth is 2 ft? a. 5 in/min b. 6 ft/s c. 3 in/min d. 5 ft/s

20. The height of a right circular cylinder is fixed at 6 cm. How fast is the volume changing at the instant when r = 3 cm, if the radius increases?

a. 3

20 cmcm

pi b. 3

36 cmcm

pi c. 3

2 cmcm

pi d. 3

12 cmcm

pi

21. If s = t2 t3, the velocity when the acceleration is zero a. v = 1/3 b. v = c. v = 1/5 d. v = 1/6

22. The area of an isosceles triangle decreases at a rate of 12 m2/s. IF the base is always twice the altitude, how fast is the altitude changing with time when the base is 3 m. a. 4 m/s b. 8 m/s c. 3 m/s d. 5 m/s

23. Find the area of the largest rectangle that can be inscribed in a given circle of radius a cm.

a. 22 a b. 2a2 c. a2 d. 2 a

24. A lot has the form of a right triangle with perpendicular sides 60 ft and 80 ft. Find the length and the width of the largest rectangular building that can be erected facing the hypotenuse of the triangle a. 36 x 75 b. 45 x 60 c. 50 x 24 d. 50 x 100

25. Two vertical poles, respectively, 1 m and 9 m high are 6 m apart. How far from the foot of the shorter pole is the point where the line segment joining the tops of the poles subtends the greatest angles? a. 3 b. 9 c. 6 d. 5

26. The collection of all points in the plane equidistant from two fixed points is a. ellipse b. parabola c. line d. circle

27. Find 3

12 lim

2 x-

-x-

x +

a. 0 b. 1 c. d. infinite

28. If a + b = 9, find the maximum value of ab2 a. 0 b. 64 c. 98 d. 108

29. Find a point on the curve y =2e2x whose tangent line is perpendicular to the line 2x 4y = 1 a. (0,1) b. (ln 2 , 1) c. (ln 2, 4) d. (l, e2)

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30. Find the derivative of y = xx a. xx b. xx (lnx) c. xx( l + lnx ) d. none of the above

31. A hot air balloon leaves at a rate of 5 m/sec as seen by an observer 15 m away. How fast is the angle of elevation of the line of sight increasing after 4 sec? a. 0.12 rad/s b. 0.08 rad/s c. 0.03 rad/s d. 0.06 rad/s

32. Water is flowing into a conical container at a rate of 24 cm3/sec. The vessel is 18 cm deep with a radius of 6 cm. At what rate is the level of liquid rising when the water in the vessel is 3 cm deep? a. 6.28 b. 2.55 c. 4.57 d. 5.73

33. The function, 2

12 +

=

xx

xy is discontinuous at x equal to

a. 2 b. 1 c. 0 d. 2

34. Evaluate 3

029 dxx

a. 1.75pi b. 2.25pi c. 3.50pi d. 4.25pi

35. Evaluate ( ) +3

12

943 dxxx

a. 8 b. 4 c. 2 d. 6

36. Find the area of the region bounded by the lines x + y = 0, x + 2 = 0 ; y = 1/x2 a. 1 b. 2 c. 3 d. 0.5

37. Find the volume generated by rotating the region bounded by x = 2y, x 4 = 0 and y2 = 4x, about the x-axis. a. 19 b. 31 c. 52 d. 84

38. The differential equation xydxdy

dxyd 2cos232

2=++ can be classified as

a. exact b. variable separable

c. linear and homogeneous d. linear but not homogenous

39. The general solution of the ordinary differential equation xyxdxdy

2= with c = constant is

a. ( ) cxy +=221ln

b. ( ) cxy +=

221ln

2

c. 2

12 xcey +=

d. ( ) cxy +=

221ln 1

2

40. Solve the differential equation yxdxdy 26= subject to 9=y when 2=x

a. ( )23 11+= xy b. ( )52 3 = xy c. ( )23 112 += xy d. ( )23 5+= xy

41. Solve xydxdy

x = 6

a. 56 cxxy += b. 65 cxxy += c. 65 cxxy += d. 56 cxxy +=

42. Solve the homogenous equation ( ) 02 22 =++ xydydxyx a. ( ) cyxx =+ 22 3 b. ( ) cyxx =+ 222 3 c. ( ) cyxx =+ 222 3 d. ( ) cyxx =+ 22 2

43. Solve ( ) ( ) 0 32 6 2 =++ dyyxydxyx a. 023 322 =+ yxyx b. cyxyx =++ 3223 c. cyxyx =+ 3223 d. cyyyx =++ 33 22

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44. Solve ( ) 0 cos sin21 2 ==+ dyyxdxyx a. cyxx =+ sin2 2 b. 3sin 3 = cxy c. cyxx = sin2 d. 3sin 3 = cxyx

45. A certain radioactive substance decomposes at a rate proportional to the amount present. Suppose that in 25 years approximately 1.2% of the substance will disintegrate, how many per cent will remain after 100 years? a. 4.72% b. 98.8% c. 95.28% d. 25%

46. A tank initially filled with 100 gal of salt solution containing 1 lb of salt per gallon. Fresh brine containing 2 lbs of salt/gal runs into the tank at the rate of 5 gal/min, and the mixture, assumed uniform, runs out at the same rate. At what time will the concentration of the salt in the tank becomes a. 14 min. b. 28 min. c. 32 min d. 55 min.

47. The Laplace transform of t is a. 1/s y b. 2/s2 c. 1/s2 d. 1/s-1

48. The Laplace transform of et is a. 1/s b. 1/(s + 1) c. 1/(s-1) d. 1/(s-1)2

49. Which of the following is a differential equation of the first order and of degree one? a. (y)3 + 2y= 3 b. dy/dx + (9-x)/x = y3

c. 2Q/x2 Q/y = 0 d. (dy/dx)2 = y + x

50. The differential equation given is correctly described by which one of the following choices: d2y/dx2 + dy/dx + bxy = f (x) a. linear, second order, homogenous b. non linear, second order, homogenous

c. linear, second order, non homogenous d. non-linear, second order, non homogenous

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CENTRAL COLLEGES OF THE PHILIPPINESCENTRAL COLLEGES OF THE PHILIPPINESCENTRAL COLLEGES OF THE PHILIPPINESCENTRAL COLLEGES OF THE PHILIPPINES College of Engineering

INTEGRATIVE ENGINEERINGINTEGRATIVE ENGINEERINGINTEGRATIVE ENGINEERINGINTEGRATIVE ENGINEERING

REMEDIAL EXAMREMEDIAL EXAMREMEDIAL EXAMREMEDIAL EXAM MATHEMATICSMATHEMATICSMATHEMATICSMATHEMATICS

(PART 2)(PART 2)(PART 2)(PART 2)

NAME: DATE SCORE i

ANSWER SANSWER SANSWER SANSWER SHEETHEETHEETHEET

A B C D A B C D 1. O O O O 26. O O O O 2. O O O O 27. O O O O 3. O O O O 28. O O O O 4. O O O O 29. O O O O 5. O O O O 30. O O O O 6. O O O O 31. O O O O 7. O O O O 32. O O O O 8. O O O O 33. O O O O 9. O O O O 34. O O O O 10. O O O O 35. O O O O 11. O O O O 36. O O O O 12. O O O O 37. O O O O 13. O O O O 38. O O O O 14. O O O O 39. O O O O 15. O O O O 40. O O O O 16. O O O O 41. O O O O 17. O O O O 42. O O O O 18. O O O O 43. O O O O 19. O O O O 44. O O O O 20. O O O O 45. O O O O 21. O O O O 46. O O O O 22. O O O O 47. O O O O 23. O O O O 48. O O O O 24. O O O O 49. O O O O 25. O O O O 50. O O O O