Mechanical Engineering Review Online

1. Find the derivative of the (3 - x²)^0.5

a. x/(3-x²)^0.5

b. 2x/(3-x²)^0.5

c. x/(3x²-1)^0.5

d. -x/(1-2x²)^0.5

2. Find the derivative of the 3(x² -1)^-1

a. -6x/(x²-1)²

b. 6x/(x²-2)²

c. 8x/(x²-1)²

d. -8x/(1-x²)²

3. Find the derivative of (4-x)/(x²-2)

a. (x²-8x+2)/(x²-2)²

b. (x²+8x-2)/(x²-2)²

c. (x²+8x+2)/(x²-2)²

d. (x²-8x-2)/(x²-2)²

4. Find the derivative of 4x² + 8x + 10.

a. 7x+8

b. 7x-8

c. 8(x+3)

d. 8(x+1)

5. Evaluate: lim_Δx→0 (x³-21)/(x-4)

a. 32

b. 40

c. 48

d. 52

6. Find the radius of curvature of the equation y = 4x³ – 4x² + 4x – 2 at point (3, 6).

a. 11,312

b. 12,091

c. 10,650

d. 9,012

7. Evaluate the limits (4x-1)/(10x+7) as x approaches infinity.

a. 3/5

b. 2/5

c. 4/5

d. 5/3

8. Find the derivative of y = (1-3x²)/(x-3) at x=1.

a. 2.5

b. 4.0

c. 3.0

d. 3.5

9. Find the partial derivatives with respect to x: x² + 6x + 10z².

a. 2x+6

b. 3x+2

c. 2x-8

d. 4x+3

10. Find the partial derivatives with respect to y: 5y² – 6y + 8

a. 8y+6

b. 10y-6

c. 12y+3

d. 6y-9

11. Find the partial derivatives with respect to x: 3x² – 4xy

a. 6x+5y

b. 8x+3y

c. 6x-4y

d. 6x-5y

12. Find the equation of the line tangent to the curve y = x + 3x^1/4 at point (9, 13).

a. 8x+6y+20=0

b. 8x-7y+19=0

c. 9x+7y+21=0

d. 9x-7y+18=0

13. What is the derivative with respect to x of (x + 2)³ – x³?

a. 12(x+1)

b. 14(x-1)

c. 12(x2+2)

d. 14(x-3)

14. Differentiate y = sec(x² + 3)

a. 3xsec(x²+3)tan(x²+3) 

b. 2xsin(x²+3)cos(x²+3)

c. -3xsec(x²+3)tan(x²+3) 

d. 2xsec(x²+3)tan(x²+3) 

15. Find the derivative of f(x) = [x⁴ – (x – 1)³]³?

a. 3[x⁴-(x-2)³]²(4x³-3x²+6x-4) 

b. 3x⁴-(x+2)³]²(4x³+3x²+6x-4)

c. 3[x⁴+(x-3)³]²(4x³-5x²+6x-4)

d. 3[x⁴-(x-1)³]²(4x³-3x²+6x-4)

16. What is the slope of the graph y = -3x² at the point (1, 4)?

a. -8

b. -5

c. -6

d. -4

17. The motion of a particle along a straight line is described by the equation x(t) = t³ – 4t² – 40t + 60 where x is expressed in feet and t in seconds. Compute the acceleration of the particle at the time in which v(t) = 0.

a. 23.32 fps

b. 28.27 fps

c. 32.09 fps

d. 20.12 fps

18. The motion of a particle is described by the equation x(t) = 3t³ – 10t² – 45t + 50, where x is measured in meters and t is measured in seconds. Find the velocity when the acceleration of the particle is equal to zero.

a. -48.11 m/s

b. -52.21

c. -61.28

d. -56.11

19. What is the slope of the line tangent to the parabola y = 14x² + 4 at point where x = 2?

a. 56

b. 52

c. 58

d. 50

20. The position of the acceleration of the object as a function of time is described by: x = 4t³ + 3t² – t + 4. What acceleration of the object at t = 3?

a. 75

b. 78

c. 82

d. 68

21. At what value of y does the inflection point occur for the curve y = 4x³ – 5x² – 25x + 26?

a. 12

b. 18

c. 20

d. 15

22. At which value of y does the relative minimum occur for the curve y = 3x³ – x² – 20x + 20.

a. -3.22

b. -4.39

c. -2.27

d. -1.88

23. The sum of two positive numbers is 60. What are the numbers if their product is to be the largest possible.

a. 40 and 20

b. 35 and 25

c. 45 and 15

d. 30 and 30

24. If the radius of a circle increases at the rate of 0.2 in/sec, find the rate of change of area when the radius is 5 inches long.

a. 3π

b. 2π

c. 6π

d. 4π

25. Find the maximum area of a rectangle whose perimeter is 150 in.

a. 1,310.23 in²

b. 1,406.25 in²

c. 1,620.21 in²

d. 1,812.14 in²

26. Find the maximum area of triangle whose perimeter is 40 in.

a. 76.98 in²

b. 83.14 in²

c. 85.14 in²

d. 74.11 in²

27. A spherical balloon is being filled with a rate of 2.4 cubic foot per second. Compute the time rate of change of the surface area of the balloon at the instant when the volume is 130 ft³.

a. 2.3 ft²/sec

b. 3.2 ft²/sec

c. 1.5 ft²/sec

d. 4.8 ft²/sec

28. A particle moves according to the following functions of time: x(t) = 3sin t, y(t) = 2cos t. What is the resultant velocity at t =π?

a. 5.0

b. 4.0

c. 3.0

d. 2.0

29. Given the function f(x) = 2x³ – 4x – 6, the minimum value of the function is:

a. (3)^0.5/3

b. -(5)^0.5/3

c. (8)^0.5/3

d. (6)^0.5/3

30. The value of x which provides the minimum y in the function: y = 3x³ – 20x + 16 is:

a. 1.49

b. 2.12

c. 3.33

d. 4.18

31. What is the radius, r, and height, h, of cylindrical oil can that holds 8 liters of oil but must have minimum surface area?

a. 11.8 cm, 28.1 cm

b. 10.8 cm, 21.7 cm

c. 12.1 cm, 24.8 cm

d. 9.2 cm, 25.1 cm

32. The product of two positive numbers is 18. Find the number if the sum of one and square of the other is least.

a. 9 and 2

b. 6 and 3

c. 18 and 1

d. 12 and 6

33. Differentiate y = e^x cos x³

a. e^x(cosx³ – 3xsinx³) 

b. 2e^x(cosx³ + 3xsinx³)

c. e^x(sinx³ – 3xcosx³)

d. -e^x(2cosx³ – 3xsinx³)

Answer Key (Handwritten)