課程名稱︰微積分甲下
課程性質︰系定必修
課程教師︰黃漢水
開課學院：生機、生工、地質、工管
開課系所︰數學系
考試日期（年月日）︰2009/04/24
考試時限（分鐘）：120分鐘
是否需發放獎勵金︰是
（如未明確表示，則不予發放）
試題 :
一 Find the following integrals: (30%)
(1) 1 e^x+5 (2) +∞ x^2
∫ -------------dx ∫ -----dx
0 e^2x+3e^x+2 0 e^x
(3) 16 x^1/4
∫ -----------dx
1 1+x^1/2
e
二 Let In(x) = ∫2x(lnx)^ndx and Jn = ∫2x(lnx)^ndx (15%)
1
(1)Find relation between In(x),In(x-1)
(2)Find relation between Jn,Jn-1,
(3)Find the values J1,J2,J3,J4
三 Find the radius of convergence and the interval of convergence
of the following power series: (15%)
(1) +∞ 1 (2) +∞ n
Σ -----(x-2)^n Σ -----------x^n
n=1 n^2 n=1 2^n(n+1)
+∞
四 (1)Let f(x) = Σ Cn*x^n = C0+C1x+C2x^2+........+Cnx^n+.......
n=1
for any |x|<δ,
+∞
g(x) = Σ n*Cn*x^n = C1x+2C2x^2+3C3x^3+.......+nCnx^n+
n=1
....... for any |x|<δ,where δ>0. Find the relation between f(x)
and g(x). (8%)
+∞ n
(2) Find the value Σ ----- (7%)
n=1 5^n
五 Let r be the radius of convergence of the power series
+∞ n+2 3 -4 5
Σ (-1)^n+1-------x^n+1 = ---x^2 + ----x^3 + ---x^4 +.......+
n=1 n 1 2 3
n+2
(-1)^n+1-------x^n+1 + ......... and let g(x) =
n
+∞ n+2
Σ (-1)^n+1-------x^n+1 for any |x|<δ (15%)
n=1 n
Find the function g(x) and the values g(0.5),g'(0.3)
(6) (9) (12)
六 Let f(x) = x^3cosx+xsinx. Find the values f(0),f(0),f(0).
(10%)
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