[試題] 97下 黃維信 微積分下 期中考
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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