課程名稱︰微積分甲上 課程性質︰數學系大一必帶 課程教師︰陳榮凱 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2013/11/14 考試時限（分鐘）：180 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : The total is 110 points. (1) (30 pts) (Differential Calculus) ｄ -1 (a) (5 pts) Determine ─tanh x. dx 3 df(x) x + x u (b) (5 pts) Determine ─── with f(x) = ∫ e du. dx 0 -1 ─ 2 x (c) (10 pts) Prove that f(x) = e has derivative and second derivative 0 at x = 0. 2 3x (d) (10 pts) Consider the function f(x) = ──. Sketch the graph of f(x) by 2 x -1 locating all relative maxima, minima, inflection points, non- differentiable points if any. Also determine lim f(x) and lim f(x). x→∞ x→-∞ (2) (30 pts) (Integral Calculus) π/2 (a) (5 pts) Evaluate ∫ sec x tan x dx. π/3 4x (b) (10 pts) Determine ∫────── dx. 2 4 (x -1)(x +1) 1 n-1 m-1 (n-1)!(m-1)! (c) (10 pts) Show that ∫ x (1-x) dx = ────── for any positive 0 (n+m-1)! integer n, m. This is the so-called β function. ∞ １ (d) (5 pts) Determine the convergence of ∫ ───── dx. 2 3 x(log x) (3) (20 pts) (Continuity) 2 (a) (10 pts) Show that f(x) = x + 1 is not uniformly continuous on R. (b) (10 pts) Prove that f(x) is continuous at x if and only if lim f(x ) 0 x →x n n 0 = f(x ) for any sequence {x } converging to x . 0 n 0 (4) (10 pts) For any a＜b＜c＜d, prove that there exists a differentiable function f(x) on R such that f(x) = 1 if xε[b,c] and f(x) = 0 if x≧d or x≦a. (5) (10 pts) Let f(x) be a differentiable function on R. Suppose that f(0) = 0 and |f'(x)|≦|f(x)|. Prove that f(x) = 0 identically. (6) (10 pts) For any n≧2εN, let v(n) be the number of prime factors of n. For v(n) example, v(4) = v(5) = 1, v(6) = 2, v(30) = 3. Prove that lim ── = 0. n→∞ ｎ //ε是屬於符號 --

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