114年 - 114 國立臺灣大學_轉學生招生考試_數學系:微積分(A)#137744

1.Determine whether converges or diverges. x()?nia( + ) xP()

2.Find the least value a such that (14)1)+a ＞e for everyx ＞ 0.

(a)  For every x € [0,27], determine limn0 fn(x).

(b)  Is the convergence in part (a) uniform on [0, 27]? Justify your answer.

4. Fix positive numbers a,b, c. For any (x, y, z) on the ellipsoid x²/a² + y²/b² + z²/c² = 1 with x ＞ 0, y ＞0,z ＞0, let A(x, y, z), B(x, y,z) and C(x, y, z) be the intercept that the tangent plane of the ellipsoid at (x,y, z) makes on the x-axis, the y-axis and the z-axis, respectively. Find the minimum of A(x,y, z) + B(x, y, z) +C(x,y, z).

5. Let U be the region in R² defined by r ≤1+ cos in polar coordinate. Let y be the boundary of U; namely, y is given by r = 1 + cos 0 in polar coordinate. Does there exist a second continuously differentiable function f(x,y) on R² such that fze fze 2x?+ Jy2 = 2x+ 3y² at every (x, y) e U,and () = (4x2-3y²+x,y³ –6y) atevery (x,y) €y? Ifyour answer is YES, construct such a function. If your answer is NO, give a proof.

6.Let Σ be the part of the paraboloid z = x²+y² that lies within the cylinder x² + y² = 1. Let n be its upward unit normal field. For the vector field F = (2x + e + z², In(z + 2) - y, 2), compute the flux integral //F.nds.