課程名稱︰微積分乙
課程性質︰必帶
課程教師︰余正道
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2016/11/10
考試時限（分鐘）：160
試題 :
There are seven problems 1~7 in total; some problems contain sub-problems,
indexed by (a), (b), etc. Notice that in the following, ㏒ x = ㏒_e x =ln x.
1.[25%] Compute the following. ( (e) is a function, and others are numbers.)
(a)
lim (1/n) [ 三次根號(1/n) + 三次根號(2/n)+....+ 三次根號(n/n) ]
n→∞
(b)
log x
lim ────────────
x→1^{-} | x-1 |
(c) lim (√(x^2+3x-5) - x ) 註:根號是整個x^2 + 3x - 5
x→∞
(d) log( x + √x ) - log 6
lim ────────────
x→4 x-4
(e) d/dx ( f(x)^{u(x)} ) ( f(x) > 0 )
2.[15%] Find the indefine integrals ∫f(x) dx in the following cases.
(a) f(x) = sin (三次根號(x) )
(b) f(x) = 1/(x^2 + x√x)
3.[10%] A painting has heigt h and is hung so its lower edge is of the
distance d above the eye of an observer. How far from the wall should
the observer stand to maximize the angle θ?
(圖略)
4.[20%] Sketch the graph of the function
-2x^2 + 5x - 1
f(x) = ────────
2x - 1
Indicate the intervals where f(x) is increasing / decreasing and concave
up/down, the critical/ inflection points and the asymptotes.
5.[10%] Compute the arc length of the curve y = (e^x + e^{-x})/2 from
x = -1 to x = 1.
6.Consider f(x) = 2x^5 - 3x^3 + 4x
(a)[5%] Show that f(x) is strictly increasing, meaning: if a < b,
show that f(a) < f(b).
(b)[10%] Let g(x) be the inverse function of f(x). Compute g'(3) and
3
∫ g(x) dx. (Notice that 3 = f(1), 0 = f(0).)
0
7.Consider the fuction
∞
Γ(α) = ∫ t^(α-1)e^(-t) dt
(a)[5%] Suppose α>0 is fixed. Show that the two improper integrals
1 ∞
∫ t^(α-1)e^(-t)dt, ∫ t^(α-1)e^(-t) dt
0 1
exist.
(b)[10%] Let n be a positive integer. Show that Γ(n+1) = nΓ(n) and
Γ(n) = (n-1)!