101年 - 101 國立中正大學_學士班二年級轉學生招生考試試題_全校：微積分#122384

2. Find the interval of convergence of the series __(b)__

3. For what value of C does the equation ln x = Cx³ have exactly one solution? __(c)__

4. Find the linearization L(x, y) = Ax + By + C of the function f(x,y) = at the point (-5,4). __(d)__

5. Find the arc length of the curve y =, x ∈ [0, 1]. __(e)__

6. The base of a solid is the ellipse x² + 4y² = 4, and every parallel cross-section perpendicular to the x-axis are equilateral triangles (等邊三角形). Find the volume of the solid. __(f)__

7. Find the area of the region enclosed by the curve r² = 4 sin 2θ. __(g)__

8. Let S be the surface of the solid E that lies above the cone z = and below the sphere x² + y² + z² = 2, and F(x, y, z) = (z, y, sin(x + y)). Evaluate ∬ₛ F•dS = __(h)__

(a) ∫∫∫ f(x,y,z)dzdzdy

(b) ∫∫∫ f(x,y,z)dydzdx

(a) First of all, compute the double integraldA

(b) Apply Fubini's theorem and symmetric property of the function e⁻ˣ² to find the integral dx:

(a) Derive the formula: xⁿ(lnx)^kdx = xⁿ(lnx)^k-1dx

(b) Derive the formulaxⁿ(lnx)ⁿdx =

(c) Use the formula x^x = e^xlnx and the Maclaurin series for eˣ to derive the formula

4. (10分) Let f(x,y) = 8xy - 2x - 4y + 5. Find the absolute minimum value of the function f(x, y) on the set D, where D is the region bounded by the parabola y = x² and the line y = 4.

101年 - 101 國立中正大學_學士班二年級轉學生招生考試試題_全校：微積分#122384

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