课程名称︰微积分甲上 课程性质︰数学系大一必带 课程教师︰陈荣凯 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2014/1/7 考试时限（分钟）：180 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : The total is 105 points. (1) (15 pts) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. n ∞ (-1) (a) Σ ──── n＝2 2 (log n) 2 ∞ n + 2 n (b) Σ ( ─── ) n＝1 2 3n + 1 ∞ π (c) Σ sin ─ n＝1 ｎ (2) (10 pts) Evaluate the following limits: sin x - x (a) lim ───── x→0 3 x x (b) lim x x→0+ 2 (3) (10 pts) Consider f(x) = x in -π＜x＜π. Determine its Fourier series. sint ╭ ── , t≠0 x (4) (10 pts) Consider f(t) = │ ｔ . Let Si(x) =∫f(t)dt. Estimate ╰ 1, t＝0 0 π Si(─) by using Simpson's Rule with n = 4 and estimate the error. ２ (5) (10 pts) Consider the cycloid given by x(t) = r(t - sin t) and y(t) = r(1 - cos t). Find its arc length for 0≦t≦2π. １ (6) (20 pts) Find the Taylor series of f(x) = ─── at x = 0. Determine its ____ √1-x 1 radius of convergence. Also find the Taylor series of f(x) at x = - and 2 determine its radius of convergence. (7) (10 pts) Let θ, θ, θ be the angles of a triangle. Prove that 1 2 3 _ 3√3 sinθ + sinθ + sinθ ≦ ── . 1 2 3 ２ (8) (20 pts) Suppose that f (x) converges to f(x) in the interval [a,b]. n (a) Give an example that f (x) is continuous in [a,b] for all n but f(x) is n not continuous. (b) If f (x) converges to f(x) in the interval [a,b] uniformly, prove that n f (x) is continuous in [a,b] for all n implies that f(x) is continuous. n (c) Suppose that f (x) is continuous in [a,b] for all n and converges to n b 2 b 2 f(x) uniformly, then ∫f (x) dx converges to ∫f(x) dx. a n a -- 第01话 似乎在课堂上听过的样子 第02话 那真是太令人绝望了 第03话 已经没什麽好期望了 第04话 被当、21都是存在的 第05话 怎麽可能会all pass 第06话 这考卷绝对有问题啊 第07话 你能面对真正的分数吗 第08话 我，真是个笨蛋 第09话 这样成绩，教授绝不会让我过的 第10话 再也不依靠考古题 第11话 最後留下的补考 第12话 我最爱的学分 --

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