107 年 - 107 台灣綜合大學系統學士班轉學生聯合招生試題：微積分A#96949　

(a) (5 Points)

(b)(5 Points)

2. (10 Points) Find the point on the curve y = that is closest to the point .

3. (10 Points) Find the length of the parametric curve
x=3t²,y=2t³, 0 ≤ t ≤ 1

4. (10 Points) Determine whether the improper integral
  
is convergent or divergent.

5. (10 Points) Find the radius of convergence of the power series 
 and evaluate 

6.(10 Points) Find the equation of the tangent plane and the normal line to the surface

at (1,1,0).

7.(10 Points) Let f : be a function. Assume that all second partial derivatives of f exist and continuous and

 Deine g :

8. (10 Points) Evaluate the double integral , sin(x+y)dA where D is the regsion bounded by x+y= π, x+y = 0 and x -y =π and x-y=0.

9.(10 Poins) Evaluate the triple integral , where E is the solid region in bounded by the surface z = 1 and z = x²+y².

10. (10 Points) Evallate the line interal along the curve C : r(t) = (t²+1)i+ (t² - 1)j, 0 ≤ t ≤ 2. Hint you may find a potential function f of the vector field F(x,y) = find f so that F = ∇f.