課程名稱︰微乙 課程性質︰必修 課程教師︰張秀瑜 開課學院：理學院 開課系所︰經濟／管院／地理 考試日期（年月日）︰2011/4 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是~ （如未明確表示，則不予發放） 試題 : 1.(16%)Evaluate the integral 2 x^2+1 (a)∫________ dx 1 x^2+x 3x+1 (b)∫___________ dx x^2+2x+5 2.(14%) Evaluate the improper integral ∞ (a)∫ xe^-xλ dx , λis positive 0 ∞ (b)∫ (x^2)e^-xλ dx , λis positive 0 3.(16%) (a)Find the radius and the interval of convergence of the series ∞ (x+1)^n Σ ------- n=0 n2^n ∞ ∞ (b)if ΣCn2^n is convergent, does it follow that ΣCn√3^n is convergent? n=0 n=0 4.(18%) Determine the series for convergence or divergence ∞ lnx ∞ 1 ∞ (a) Σ ------ (b) Σ ------------- (c) Σ(-1)^n arctan(n) n=1 e^n n=1 (n^4+1)^1/3 n=1 5.(12%) Find the Tylor polynomial T3(x) of f(x)=sinx at a=π/2 and use Tylor's formula tp express the reminder term R3(x) 6.(8%) Use the binomial series to find f(6)(0) of f(x)=(x^2+4)^1/2 ↑上標 7.(16%) (a) Find the area of the region that lies in the sector 1 r=---- π＜θ＜2π θ (b)For the curve in (a), find the slope of the tangent line when π=θ --

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