課程名稱︰微積分下 課程性質︰系必帶 課程教師︰黃維信 開課學院：工學院 開課系所︰工科海洋系 考試日期（年月日）︰2009/4/22 考試時限（分鐘）：120 是否需發放獎勵金：是 試題 : 1.State whether the sequence or the expression converges and if it does,find the limit.(15%) 5n+2ln(n) (a)lim ──── n→∞ n+3ln(n) arcsinx (b)lim ─── x→0 x 1 (c)lim x^(──) x→1 x-1 2.Determine whether the integral converges and,if so,evalutate the integral. (10%) π/2 (a)∫ secxdx 0 ∞ dx (b)∫ ──── 2 x(lnx)^k 3.Determine whether the series converges or diverges(10%) ∞ 1 (a)Σ ──── k=1 kln(k+1) 2×3 5×6) (b)── - ── +...(-1)^k[(3k+2)(3k+3)/(3k+4)(3k+5)] 4×5 7×8 4.Expand sinx in powers of x-π and specify the values of x for which the expansion is valid(15%) 5.Show the principal normal vector is perpendicular to the tangent vector of a space curve.(10%) 6.Find the unit tangent,the principal normal, the curvature and write an equation in x,y,z for the osculating plane at the point on the curve → → → → r(t)=e^t i +e^(-t) j -t(√2) k at t=0 and find the length from t=0 to t=ln3 (25%) 7.The cylinder y^2+z-4=0 and the paraboloid x^2+3y^2=z intersect in a space curve.Determine the projection of the curve onto the xy-plane.(5%) 8.Let g be a differentiable function of one variable and set f(x,y)=g(x,y) Show that x(δf/δx)-y(δf/δy)=0 (10%) (δ為偏微分符號) --

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