90年 - 90 國立清華大學轉學生招生考試試題_八系聯招:微積分#114280

(I). 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).Let a₁≥a₂≥...≥a₁₀>0. Find . Ans.__ Z__

(3). Evajuate the integral dx. Ans.__丙__

(4).Evaluate the itegral dydx. Ans. __丁__

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

(1). Let f be a C² function on R³. Show that if w a f(r-s,s -t,t -r), then

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

(3). Let , be a sequence with nonnegative ternus. Suppose that converges. Does convenge? 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.