96年 - [非官方正解]96 國立暨南大學轉學生招生考試試題_微積分_(財金系)#125242

1. Find the area of the region bounded by the graph of , the x-axis, and the line x=3. (A)(B)(C)(D)

2. Find the area of the region between the graphs of f(x) = 3x³ - x² - 10x and g(x) = -x² + 2x. (A) 0 (B)12 (C)18 (D)24

3. Evaluate ? (A) (B) (C) (D)

4. Find the arc length of the graph of f(x) = from x = 0 to x = 1. (A)(B)(C) (D)

5. Find the general solution of

 y' + xy = , where C stands for constant. (A) (B) (C)(D)

6. Find the general solution of

y' - y tan t = 1, , where C stands for constant. (A) y = sec t + C cos t (B)y = tan t + C sec t (C)y = sec t + C sin t (D)y = tan t + C cos t

7. Evaluate (A) π (B) (C) (D)

8. Find the distance between the two parallel planes given by 3x - y + 2z - 6 = 0 and 6x - 2y + 4z + 4 = 0 (A) (B) (C) (D)

9. Find the distance between the point (3, -1, 4) and the line given by x = -2 + 3t, y = -2t and z = 1 + 4t. (A) √3 (B)2√2 (C)√6 (D)4√3

10. Find the minimum value of f(x, y, z) = 2x² + y² + 3z² subject to the constraint 2x - 3y - 4z = 49. (A) 156 (B)124 (C)147 (D)136.

11. Find the maximum value of √(x²)/9 + √(y²)/16 = 1 , where x ＞ 0 and y ＞ 0, subject to the constraint = 1. (A) 12 (B)18 (C)24 (D)36

12. Find the volume of the solid region bounded by the paraboloid z = 4 - x² - 2y² and the xy-plane. (A) 2√2π (B)4√2π (C)6√2π (D)8√2π

13. Evaluate sin (y²)dz dy dx (A) 1 (B)1/2 (C)2/3 (D)3/4

14. Let R be the region bounded by the lines x - 2y = 0, x - 2y = -4, x + y = 4, and x + y = 1. Evaluate the double integral dA (A) 164/3 (B)164/5 (C)164/7 (D)164/9

15. Let R be the region bounded by the square with vertices (0,1), (1,2), (2,1), and (1,0). Evaluate the integral (x + y)² sin²(x−y) dA R (A) (B) (C) (D)

16. Evaluate the surface integral

  ds S

where S is the first-octant portion of the plane

2x + y + 2z = 6 (A) 73/2 (B)73/4 (C)243/2 (D)243/4

17. Find the series representation of the function defined by the integral

 (A) (B) (C) (D)

18. The production function for a candy manufacturer is f(x, y) = 4x + xy + 2√y where x is the number of units of labor and y is the number of units of capital. Assume that the total amount available for labor and capital is 2000, and that units of labor and capital cost 20 and 4, respectively. Find the maximum production level for this manufacturer. (A) 13,844.2 (B)12,425.5 (C)13,201.8 (D)11,392.7

19. Find the angle θ between the vectors u =＜ 1,1,1＞ and v =＜ 2,1,-1＞ (A) (B) (C) (D)

20. Determine whether u and v are orthogonal, parallel, or neither u =＜ 4,3 ＞, v =＜ ＞ (A) Orthogonal but not parallel (B)Orthogonal and parallel (C)Parallel but not orthogonal (D)Not Parallel and not orthogonal