90年 - 90 國立清華大學轉學生招生考試試題_原子科學系:微積分#114282

(1). Let f(x) = .Which of the following is true? Ans.__甲__

 a. f(1)>0    b.f(5)＜0    c.f(2)＞f(4)   d.f(2)＜f(4)

(2.)Find the Iimit Ans.__乙__

(3). Evaluate the integral Ans.__丙__ 

(4). Evaiuate the integral dydx. Ans.__丁__

(5). Find the area of the top half gt the region inside the cardioid r=1 + cosθ and outside the circle r = cosθ. Ans.__ 戊__

(1). Let f bea C¹function on R. Verify that z = satisfies the equation =0.

(2). Let f be a coutinuous real-valued function defned on R.Using integration by parts, prove
  

(3). Let be a sequence with nonnegative termas. Suppose that converges. 1 Does converge? Prove or disprove it. 

(4). Find ∫_Dy²dA, where D is the region bounded by the lines x - 2y = 2, x -2y = 5, 2x + 3y = 1 and 2x + 3y =3.

(5). Evaluate the line integral

where the curve C = {(x,y)|x²+y² = 1} is traversed counterclockwise.