13. Find the partial derivatives with respect to x of the function: xy2 – 5y + 6.A. y2 – 5B. xy – 5yC. y2

D. 2xy

14. 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)

15. Find the slope of the line tangent to the curve y = x3 – 2x + 1 at x = 1.A. 1B. ½C. 1/3D. ¼

16. Find the slope of the tangent to the curve y2 = 3x2 + 4 through point (-2, 4)

A. -3/2B. 3/2C. 2/3D. -2/3

17. Find the slope of the line whose parametric equations are x = 4t + 6 and y = t – 1.

A. -4B. ¼C. 4D. -1/4

18. What is the slope of the curve x2 + y2 – 6x + 10y + 5 = 0 at (1, 0).A. 2/5B. 5/2C. -2/5D. -5/2

19. Find the slope of the curve y = 6(4 + x) ½ at (0, 12).A. 0.67B. 1.5C. 1.33D. 0.75

20. Find the acute angle that the curve y = 1 – 3x2 cut the x-axis.A. 77°B. 75°C. 79°D. 120°

21. 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°

22. Find the equation of the tangent to the curve y = x + 2x1/3 through point (8, 12)

A. 7x – 6y + 14 = 0B. 8x + 5y + 21 = 0C. 5x – 6y – 15 = 0D. 3x – 2y – 1 = 0

23. What is the radius of curvature at point (1, 2) of the curve 4x – y2 = 0?

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A. 6.21B. 5.21C. 5.66D. 6.66

24. A function is given below, what x value maximizes y?y2 + y + x2 – 2x = 5

A. 2.23B. -1C. 5D. 1

25. The number of newspaper copies distributed is given by C = 50 t2 – 200 t + 10000, where t is in years. Find the minimum number of copies distributed from 1995 to 2002.

A. 9850B. 9800C. 10200D. 7500

26. Given the following profit-versus-production function for a certain commodity:

P = 200000 – x – (1.1

1+ x) 8

Where P is the profit and x is unit of production. Determine the maximum profit.A. 190000B. 200000C. 250000D. 550000

27. The cost C of a product is a function of the quantity x of the product is given by the relation: C(x) = x2 – 4000x + 50. Find the quantity for which the cost is a minimum.

A. 3000B. 2000C. 1000D. 1500

28. If y = x to the 3rd power – 3x. find the maximum value of y.A. 0B. -1C. 1D. 2

29. Divide 120 into two parts so that product of one and the square of the other is maximum. Find the numbers.

A. 60 & 60B. 100 & 20C. 70 & 50D. 80 & 40

30. If the sum of two numbers is C, find the minimum value of the sum of their squares.

A. C2 / 2B. C2 / 4C. C2 / 6D. C2 / 8

31. A certain travel agency offered a tour that will cost each person P 1500.00 if not more than 150 persons will join, however the cost per person will be reduced by P 5.00 per person in excess of 150. How many persons will make the profit a maximum?

A. 75

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B. 150C. 225D. 250

32. The shortest distance from the point (5, 10) to the curve x2 = 12y is:A. 4.331B. 3.474C. 5.127D. 6.445

33. 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 mB. 3.51 mC. 3.71 mD. 4.41 m

34. An iron bar 20 m long is bent to form a closed plane area. What is the largest area possible?

A. 21.56 square meterB. 25.68 square meterC. 28.56 square meterD. 31.83 square meter

35. A Norman window is in the shape of a rectangle surmounted by a semi-circle. 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. 1B. 2/3C. 1/3D. ½

36. 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 parallel to one side?

A. 65,200B. 62,500C. 64,500D. 63,500

37. An open top rectangular tank with square bases is to have a volume of 10 cubic meters. The material for its bottom cost P 150.00 per square meter, and that for the sides is P 60.00 per square meter. The most economical height is:

A. 2 metersB. 2.5 metersC. 3 metersD. 3.5 meters

38. 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 cmB. 32.28 cmC. 18.41 cmD. 28.74 cm

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