TECHNOLOGICAL INSTITUTE OF THE PHILIPPINESMANILADIFFRENTIAL
CALCULUS COMPETENCY EXAMSTUDENT
NAME___________________________________DATE_____________________________DIRECTIONS:
PROVIDE THE COMPLETE STEP BY STEP SOLUTIONS TO SELECT THE BEST
ANSWER.1. A right triangle has a fixed hypotenuse of 30 cm. And the
other two sides are allowed to vary. Determine the largest possible
area of the triangle.a. 225 sq.cmb. 234 sq.cmc. 243 sq.cmd. 216
sq.cm 2. Find the area of the largest rectangle that can be
inscribed in a semicircle of radius r.a. 2r2b. r2c. r2d. r2
3. An open box is made from a square piece of cardboard (of side
1) by cutting out four equal (small squares) at the corners and
then folding. How big should the small squares be in order that the
volume of the box be as large as possible?a. 1/8b. 1/10c. 1/6 d.
NOTG
4. A boy 5 feet tall walks at a rate of 3 feet/sec toward a
streetlamp that is 12 feet above the ground. a) What is the rate of
change of the tip of his shadow?a. 15/7 fpsb. 36/7 fpsc. 12/5 fpsd.
7/15 fps
5. A conical tank 20 feet in diameter and 30 feet tall (with
vertex down) leaks water at a rate of 5 cubic feet per hour. At
what rate is the water level dropping when the water is 15 feet
deep?a. 1/9b. 1/5 c. 3/7d. 4/7
6. The dimension of a box are a, a 1, a + 4 inches, Find how
fast the total surface area S increases as a increases.a. 15(a +
1)b. 12(a + 1)c. 15(a 1)d. 12(a 1)
7. Find the minimum of the function f(x) = 2x3 + 3x2 12x + 17.a.
(1, 10)b (-1, 5) c. (-2, 10) d.(-2, 37)
8. A farmer is to make a rectangular paddock. The farmer has 100
metres of fencing and wants to make the rectangle that will enclose
the greatest area. What dimensions should the rectangle be?a. 25 x
25b. 10 x 30c. 15 x 35d. 16 x 34
9. What is the radius of an expanding circle at the moment when
the rate of change of its area is numerically twice as large as the
rate of change of its radius?
a. b. c. d.
10. If the parabola y=x2+C is tangent to the line y=4x+3, find
the value of C.a. 4b. 7c. 6d. 5
11. A rectangular yard is to be built which encloses 400 sq. ft.
Two opposite sides are to be made from fencing which costs 10 peso
per foot, while the opposite sides are made from fencing which cost
20 peso per foot. Find the least possible cost?a. 1031.3 pesob.
1131.3 pesoc. 1231.3 pesod. 1331.3 peso
12. The volume of a cube is increasing at the rate of 6 cm3/min.
How fast is the surface area is increasing when the length of the
edge of the cube is 12 cm? a. 3 cm2/minb. 2 cm2/minc. 2.5 cm2/mind.
3.5 cm2/min
13. A balloon leaving the ground, 18 m from the observer, rises
vertically at a steady rate of 3 m/s. How fast is the angle of
elevation of the line of sight increasing after 8 s.a. 0.12 rad/sb.
0.08 rad/sc. 0.03 rad/sd. 0.06 rad/s
14. Find the point on parabola y=12x2 that is closest to the
point (-4,1).a. (2,-2)b. (-2, 2)c.(1, -1)d. (-1,1)
15. If s = t2 - t3, find the velocity when the acceleration is
zero.a. v = 1/3b.v = c. v = 1/5d.v = 1/6
16. A spherical balloon is being filled with air at the rate of
1 cfs. Compute the time rate of change of the surface area of the
balloon at the instant when its volume is 113.1 cubic feet.a. 0.67
ft2/secb. 1.73 ft2/secc. 3.0 ft2/secd. 3.7 ft2/sec
17. Find the local maxima of the function f(x) = 2x3 9x2 + 12x
5.a. x =2b. x = 1c. x = -1 d. NOTG
18. At what points does the curve y = 4x x2 have a slope of 4.a.
(0,0)b. (1,2)c. (3,2)d. (-1,2)
19. A particle moves so in a straight line according to the
equation S=t3+at2+bt. If the initial velocity (at t=0) is 5, find
the value of a such that when t = 1, it is moving with four times
its initial velocity. a. 5b. 6 c. 7d. 4
20. The height and base radius of a right circular cylinder are
20 cm and 8 cm respectively. If the height decreases at the rate of
3 cm/s and its base radius increases at the rate of 2 cm/s, at what
rate is its volume changing?a.448 pib.428 pi c.438 pi d. 418 pi
