課程名稱︰微積分甲上 課程性質︰暑修 課程教師︰朱樺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰106年8月4號 考試時限（分鐘）：150分鐘 試題 : (1) (30%) Find an antiderivative (or the indefinite integral) of the function. (Need not give any explanation.) (a) x^e (b) π^x (c) 1/x (d) 1/(x^(1/5)) (e) 1/(1+x^2) (f) 1/√(1-x^2) (g) sin x (h) cot x (i) sec x (j) ln x (2) (14%) Find the limit. n^2 n^2 n^2 n^2 (a) lim (------- + ------- + ... + ------- + ... -------) n→∞ n^3+1^3 n^3+2^3 n^3+i^3 n^3+n^3 0 t-x^2 ∫ e (2t^2+1) dt x^2 (b) lim ----------------------- n→∞ x^4 (3) (21%) Evaluate the integral. 2x e^(2x) (a) ∫(2e ln x + -------) dx x 1 (b) ∫-------- dθ 4+5sinθ ∞ dx (c) ∫ ---------- 0 (x+1)√x (4) (7%) The base of the solid S is the triangular region with vertices (0,0), (1,0) and (0,1). Cross-sections perpendicular to the y-axis are of equilateral triangles. Find the volume of S. (5) (7%) Calculate the volume generated by rotating the region bounded by the curves y = ln x, y = 0 and x = 2 about the y-axis. (6) (a) (4%) Sketch the curve C: x = t^2, y = t^3/3-t, t∈ R. d^2 y (b) (7%) Find -----. dx^2 (c) (7%) Let R be the region enclosed by C. Find the area of surface of the solid obtained by rotating R about the x-axis. (7) (4%) Sketch the polar curve r = 1+2cos2θ. (8) (7%) Find the length of the curve r = θ^3, 0≦θ≦4. (9) (7%) Find the area of the region that lies inside the curve r = 3sinθ and outside the curve r = 1+sinθ. --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 101.139.172.18 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1502886297.A.800.html