111年 - 111 台灣綜合大學系統學士班轉學生聯合招生試題：微積分A(理工組二年級)#110842

1. (10 points) For what values of a and b, a,b≠0, is the following equation true?
  

(a) (5 points) Let f(x) = sec² x on be the inverse function of f. Find ()'(4) =

(b) (5 points) Let f = f(x,y) be a differentiable function of r and y, and let x= rs,y =rts and h(r,s) = f(x,y) = f(rs,r+ s). Assume (1,2) = 2, (1, 1) =________

3. (10 points) If dt, on what interval(s) is f increasing?

4. (10 points) Let a ＞ 0 be a constant. Evaluate

5. (10 points) Find the area of the region that lies inside the curve r = 4sinθ and outside the curve r = 2.

6. (10 points) For what real values of p does the series converges?

7. (10 points) Let f(x) be the function defined by the power series

Try to express f(s) as an elementary function.

9. (10 points) Evaluate dA, where R is the region in the first quadrant bounded by lines y = x, y = 3x, and the hyperbolas xy = 1, xy = 3.

10. (10 points) Evaluate the line integral dy, where C is the arc of the the circle x² + y² = 4 traversed counter-clockwise from (2,0) to (-2,0).