105年 - 105 國立臺南大學_轉學生招生考試試題_理工聯招二年級：微積分#124196

1.=?

2. For f(x)= find all real numbers a and b such that f'(0) exists.

3. Show that .

4. Evaluate =?

5. Evaluate =?

6. Let P(1,1,0) and f(x,y)=x³ y-xy³. Please find an equation of the tangent plane to z=f(x,y) at P.

7. Evaluate the double integral , where 
D≔{(x,y)| 0≤x≤2,1≤y≤2}.

8. Evaluate +xdy, where C is a line segment from (-5,-3) to (0,2).

9. Let E≔{(x,y,z)| x²+y² ≤ z ≤ 1} with the boundary ∂Eand n be the outward-pointing normal vector. Given a vector field
F(x,y,z)=(+2x-sinz,x-cosz,sin sin⁡(y²)-z²).
Please evaluate

10. Determine if the series converges. Prove or disprove it.