May 6, 2013
師大數學系轉系、輔系考試試題
說明:
• 本試卷共 10 題,每題 10 分。滿分為 100 分。
• 試題中,R 代表實數系。
1. 找出兩整數 a, b 使得 47a− 56b = 1。
2. 解下列對數不等式:
12(log2√
x)2− 7log2x− 10 > 0.
3. 坐標平面中有一等腰三角形 ABC,其中 AB = AC = 5,BC = 6。設點 H 為△ABC 的垂 心,將−−−−⇀AH 寫成−−−⇀AB 與−−−⇀AC 的線性組合−−−−⇀AH = x−−−⇀AB + y−−−⇀AC 時,兩實數 x, y 分別等於多少?
4. {1,2,3,4,5,6,7,8,9,10} 的所有子集合中,有多少個所包含的奇數個數比所包含的偶數 個數多?
5. 設矩陣 A =
[ 1 1 0 3
]
。若 A6=
[ a b c d
]
,則 b 之值為何?
6. Let f :R → R be the function given by f (x) = 3x5− 10x4+ 7. Find the point(s) of inflection of the graph of f .
7. Let f :R → R be a continuous function. If
∫ 2x
0
t f (t) dt = sin x− xcosx,
determine f (π/2).
8. Find the definite integral ∫
3 1
√x ln x dx.
9. Find the interval of convergence of the power series
∑∞ n=1
(−1)n+1(x− 2)n n 2n .
10. Let f :R → R be a function. A real number c is called a fixed point of f if f (c) = c.
Prove that if f is differentiable onR and f′(x) < 1 for all x∈ R, then f has at most one fixed point.
1