課程名稱︰微積分甲下 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）︰2014/06/19 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) (20 pts) Find three independent fundamental solutions to y"' - 8y = 0. Also verify the independency of three solutions. (2) (15 pts) Let R = { (x, y) | -a ≦ x ≦ a, -b ≦ y ≦ b } be a rectangular region. Consider a function f(x, y) such that both f(x, y) and f (x, y) are y continuous on R. Suppose furthermore that | f (x, y) | ≦ M, | f(x, y) | ≦ y M , aM < b and aM < 1. We can construct a sequence of functions by setting 1 1 x y = 0 and y (x) = ∫ f(t, y (t)) dt. Prove that y (x) converges to a 0 i 0 i-1 n function y(x) and y'(x) = f(x, y). → 3 2 3 y 3 y (3) (10 pts) Let F = (5x + 12xy , y + e sin z, 5z + e cos z). Find the → outward flux of F across the boundary of E, where E is the solid given by 1 2 2 2 ≦ x + y + z ≦ 4. 2 (4) (10 pts) Consider R := { a ≦ x ≦ b, ψ(x) ≦ y ≦ψ(x) } ⊂ R , where 1 2 ψ(x) ≦ ψ(x) are continuous function on [a, b]. Show that R is closed. 1 2 / That is, for any P ∈ R, there exists a δ > 0 such that if | Q - P | < δ / / then Q ∈ R. / → 2 (5) (10 pts) Verify Divergence Theorem for F = (x , xy + z, z), and E is the 2 2 solid bounded by the paraboloid z = 4 - x - y that lies above z = 2. (6) (15 pts) Consider the cardioid r = 2 ( 1 - sinθ) with 0 ≦θ≦ 2π. Determine the length of the cardioid and the area enclosed by the cardioid. (7) (20 pts) Let L = P dx + Q dy + R dz be a closed differentiable 1-form on 3 R . Show that L = df for some function f. (8) (10 pts) Solve the equation xy' + y = 2x. -- 第01話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了 第03話 已經沒什麼好期望了 第04話 被當、21都是存在的 第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊 第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋 第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題 第11話 最後留下的補考 第12話 我最愛的學分 --

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