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.