100年 - 100 國立高雄大學轉學生招生考試試題_電機工程學系：微積分#115945

科目：研究所、轉學考(插大)-微積分 | 年份：100年 | 選擇題數：0 | 申論題數：21

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (21)

(a)

(b) y = e^x sin x

(c)

(d) y=3^-x

(e) y=xlnx

(a)

(b) ∫ (sin x )³ (cos x )²dx

(c)

(d) ∫ x² e^x dx

(e)

3. Find the area of the region between the graphs of f (x)=3x³-x²−10x and . g(x)=-x²+2x

4. The region bounded by the parabola y = x² and the line y = 2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid

5. Find the work done by the force field in moving an object along the curve C parametrized by

6. Find the mass of the ellipsoid Q given by 4x²+4y²+z²= 16 , lying above the xy-plane. The density at a point in the solid is proportional to the distance between the point and the xy-plane (ρ = kz ) .

(a) Sketch a graph of the polar equation r = 3cos(3θ) . (4%)

(b) Use a double integral to find the area enclosed by the graph of r = 3cos(3θ) . (6%)

8. Find an equation of the tangent plane to the hyperboloid given by

z²-2x²-2y²=12
at the point (1,-1,4).

(a) Find the derivative of f (x, y,z)=x³-xy²- z at P₀(1，1，0)in the direction of . (5%)

(b) In what direction does f (x, y,z) increase most rapidly at , and what is the rate of change in this direction. (5%)

(a) Find the length of the graph of

(b) Find the length of the curve