99年 - 99 國立中山大學轉學生招生考試試題_機電系二年級：微積分#122156

1. The rate of depreciation of a machine is inversely proportional to the square of t + 2, where V is the value of the machine t years after it was purchased. The initial value of the machine was $500,000, and its value decreased $100,000 in the first year. Estimate its value after 6 years. Round your answer to the nearest integer. (A) $500,000 (B) $275,000 (C) $550,000 (D) $475,000 (E) $350,000

3. Find the cubic function of the form f(x) = ax³ + bx² + cx + d, where a ≠ 0 and the coefficients a, b, c, d are real numbers, which satisfies the conditions given below.

Relative maximum: (3,5)
Relative minimum: (5,3)
Inflection point: (4,4) (A) x³ - 6x² + x + 21 (B)x³ - 6x² - x + 21 (C) -x³ + 6x² -x + 22 (D)x³ - 6x² +x - 22 (E) x³ - 6x² + x - 22

4. Find the derivative of the function y = cosh⁻¹7x. (A) y' = (B) y' = (C) y' =-     (D) y' = (E) y' =

6. Evaluate4tan³xdx. (A) 2 - ln 2 (B) 2 + ln 2 (C) 2(1 - ln 2) (D) 2(1 + ln 2) (E) 4(1 + ln 2)

7. Find the Maclaurian series for the function f(x) = . (A) (B) (C) (D) (E)

8. Find the point on the curve y = 3x³ + 10x² + 4x at which the curvature K is zero. (A) x = - (B) x = (C) x = - (D) x = - (E) x = -

9. Find the highest point on the curve of intersection of the following surfaces: Cone: x² + y² - z² = 0, Plane: x + 14z = 10. (A) (B) (C) 130 (D) 140 (E) 150

10. Find the volume of the solid between the spheres x² + y² + z² = 25 and x² + y² + z² = 36, and inside the cone z² = x² + y². (A)(2 - ) (B) 182π(2 - ) (C) (D) 307π/3 (E) (2 - )