102年 - 102 國立高雄大學轉學生招生考試試題_電機工程學系：微積分#115829

科目：研究所、轉學考(插大)-微積分 | 年份：102年 | 選擇題數：0 | 申論題數：11

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 (11)

1. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.

(a) ∫(sin x)⁵(cos x)²dx

(b) ∫ (tan x)⁶ (secx) ⁴dx

3. A solid E lies within the cylinderx²+ y² =1, below the plane z = 4 , and above the paraboloid z=1 − x²y² . The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E.

4. Find y that satisfies

where k is a constant and y cannot exceed a maximal size M

5. Find the Maclaurin series of the functionx f(x)=e^xand its radius of convergence.

6. If f(x,y,z)=xsin(yz) , (a) find the gradient of f and (b) find the directional derivative of f at (1,3,0) in the direction of .

7. Find the work done by the force field

in moving a particle along the circular helix

from point.

8. A thin lamina has the shape of a triangle with vertices at (0,0), (2,0), and (2,4). The density function associated with the lamina has equation ρ(x, y )=4x+2y+2 . Find the mass and center of mass of the lamina.

9. Evaluate the double integral
sinθdA

where Q is the region inside the upper half of the cardioid r =1+cosθ .

10. Find the tangent plane to the surface with parametric equationsx=u² , y =v² , z= u + 2v at the point (1,1,3).