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※ 編輯: iwantowaylaw 來自: 140.112.4.94 (09/03 11:57)
課程名稱︰暑修微積分甲下
課程性質︰
課程教師︰周青松
開課學院：
開課系所︰
考試日期（年月日）︰2009/8/19
考試時限（分鐘）：8:10~10:00
是否需發放獎勵金：要
（如未明確表示，則不予發放）
試題 :
A.
(a)
ｘ^100n ∞
Determine whether the sequence { ——— } converges and if so ,
ｎ! n=1
find the limit .
(b)
∞ ｘ^k
Show that e^x= Σ —— for all real x .
k=0 ｋ！
B.
(a)
x
Given that the function f is continuous , find the limit lim (1/x) ∫ f(t)dt .
x→0 0
(b)
Show that lim x^x = 1
x→0+
C.
A non-negative function f defined on (-∞,∞) is call a probability density
∞
function if ∫ f(x)dx = 1 . And the mean of a probability density function
-∞
∞
f is defined as the number μ = ∫ xf(x)dx .
-∞
(a)
Show that the function f(x) = ke^(-kx) , if x≧0
0 , if x<0
is a probability density function where k>0 is a given constant .
This function is called the exponential density function .
(b)
Find the mean of the exponential density function.
D.
(a)
∞
Show that ∫ x^(-p) dx converges if p>1 and diverges if 0＜p≦1 .
-∞
(b)
∞
Show that Σ k^(-p) converges if and only if p>1 .
k=1
E.
(a)
Deduce the differentiation formulas
dsinhx/dx = coshx (雙曲正弦函數對x微分 = 雙曲餘弦函數)
from the expansion of sinhx and coshx in powers of x .
(b)
x
Fine a power series representation for the function ∫ (sinht/t) dt .
0
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