108年 - 108 台灣綜合大學系統學士班轉學生聯合招生試題：微積分A#96946

(a)

(b)

2.(10 points) Evaluate

3. (10 points) Given
  
with a ＞ 0, find a so that F is maximum at x = -2a.

4.(10 points) Find the largest possible area of a triangle with vertices (0, 2), (1,0) and the third vertex on the ellipse

5. (10 points) Evaluate

6. (10 points) Evaluate

7.(10 points) Derive the complete Taylor series expansion for

 about x = 0. (In the form with a general formula for )

8. (10 points) Given function T(x,y) = 1+ x²-y² , find the curve γ(t) = (x(t), y(t)) so that γ(0) = (1,4) and γ'(t) = -∇T(γ(t)).

9. (10 points) Find (a,b) with so that
ㆍThe point P = (1,a,b) is on the surface E defined by

ㆍThe tangent plane to E at P contains lines

10. (10 points) Evaluate,where
 
and L is the path from A to B, to C to A and to D as shown below:

 Note: The path from B to C is circular.