課程名稱︰微積分乙上
課程性質︰必帶
課程教師︰張樹城
開課學院：生科院
開課系所︰生科系.生技系
考試日期（年月日）︰2017/01/12
考試時限（分鐘）：110
試題 :
1.(60%) Compute the followings and justify your answers:
(a) lim (1 + 1/x)^x.
x→∞
4
(b)∫ 3x/√(1+x^2) dx.
2
4
(c)∫ 1/(x ln x) dx.
2
1
(d)∫ √(1-x^2) dx.
-1
(e)∫e^x sin x dx
(f) lim (1/n) (cos (π/n) + cos (2π/n)+...+ cos (((n-1)π)/n) + cosπ).
n→∞
1
(g)∫ ln √(x^2 + 1) dx.
0
(h)∫sin^(-1) x dx.
∞
(l)∫ e^(-x^2) dx.
-∞
4
(j)∫ e^(√x) dx.
0
(k)∫sec x dx.
∞
(l)∫ (1/x^p)dx, p > 1.
1
II.(10%)
(a) Find the solution y = y(x) of the following initial value problem:
dy/dx = y, y(0) = 1.
(b) Let f(x) be a continuous function defined by
x f(t)
f(x) = 1 + ∫ ────── dt, x > = 0.
0 (t+1)(t-2)
Find f(1).
III.(10%) The semi-circle x^2 + y^2 = r^2 is revloved about the x-axis to
generating a sphere of radius r.
(a) Find the surface area of the sphere of radius r.
(b) Find the volume of the sphere of radius r.
IV.(10%) The disk x^2 + y^2 < = a is revloved about the line x = b to
generate a solid torus with a < b.
(a) Find the surface area of the torus.
(b) Find the volume of the solid torus.
V.(10%)
Let f: [0,1] → R be a continuous function satisfying
1
∫ f(x) dx = 1/2.
0
1
(a) Define F(x) = f(x) - x, show that ∫ F(x) dx =0.
0
(b) Show that there exists c belongs to (0,1) such that f(c) = c.
∫∫∫ Happy Lunar New Year∫∫∫
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