98年 - 98 台灣聯合大學系統(清、交、陽、中四校聯招)學士班轉學生聯合招生試題：微積分#115150

1. What is the maximum value |f"(x)| for the function f(x) = x³(10 - 3x²) on the closed interval [0,2]? (A)16(B)660(C)(D)(E)360

2. If = x sin πx, what is the value of f(4)? (A)0(B)-2π (C)2π (D)(E)

3. What is the value of ? (A) (B)(C)0 (D)+∞ (E)-∞

4.Which of the following series diverges? (A)(B)(C)(D)(E)

5. What is the value of the definite integral ? (A)(B)(C)(D)e-1 (E) None of the above

6. Which function has maclaurin series ? (A) tan^-1 2x(B)ln(1+2x)(C)(D) (E) sin^-12x

7. What is the area that lies inside the circle r = 6 and above the line r = 3 cscθ? (A)(B)(C)(D)(E)

8.What is the value of the double integral dydx? (A)ln9(B)9 (C) 10 (D)ln10 (E) None of the above

9. Which of the following curve is tangent to the surface x²+ y²-z=1 when t = 1? (A)(B)(C)(D)(E)None of the above

10. Let S be the surface parametrized by r(θ, z) = 3sin 2θi + 6sin²θj+zk where 0 ≤ θ ≤ π . Which of the following is equivalent to the surface differential do? (A)dθdz (B)dθdz (C)3dθdz (D)6dθdz (E)12dθdz

1. Find the limit:. Answer :__________

2. Among all the points on the graph of parabolid z = 10 - x² - y² that above the plane x + 2y + 3z = 0, find the point farthest from the plane. Answer :_________

3. Set up the integral with the order dzdrdo for evaluating the triple integral of the function F(x, y, z) = sin(z²) over the solid region D bounded below by the plane z = 0, laterally by the circular cylinder (x - 1)² + y² = 1, and above by the paraboloid z = 2 - x²- y².

Answer :________ (Do not evaluate the integral.)

4.Use the transformation (chang of variables) x = uv^-1, y = uv to evaluate

where D =｛(x,y) :1 ≤ xy ≤4,1 ≤ y/x ≤ 4]. Answer :_________

5. At the point (1,2), the function f(x, y) has a derivative of 2 in the direction toward (2,2) and a derivative of -2 in the direction toward (1,1). Find the derivative of f at (1,2) in the direction toward the point (4,6). Answer :__________

1. Define f(x)= Show that f'(x) is continuous at x = 0.

2. Find the values a ≤ b such that dx has minimal value.

3. Let C be a simple colsed smooth curve in the plane x + 2y + 2z = 2. (Orient C to be counterclockwise when view from above.) Show that
depends only on the area of the region enclosed by C and not on the position or shape of C.

98年 - 98 台灣聯合大學系統(清、交、陽、中四校聯招)學士班轉學生聯合招生試題：微積分#115150

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