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推 ian60702 :工科海的微積分特別難= = 01/14 21:44
課程名稱︰微積分下
課程性質︰必修
課程教師︰黃維信
開課學院：工學院
開課系所︰工科海洋學系
考試日期（年月日）︰2008/4/22
考試時限（分鐘）：120mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. State whether the sequence converges and if it does, find the limit.(15%)
(a) [ln(1+1/n)]^n (b) 3ln(2n)-ln(n^3+1)
n t^2
(c) ne^(-n^2)∫ (e ) dt
0
2. Determine whether the integral converges and, if so, evaluate the
integral.(10%)
π/2 cosx ∞ dx
(a)∫ ──── dx (b) ∫───────
0 √sinx e {√(x+1)}lnx
∞
3. Show that Σ ln{(k+1)/k} diverges, although ln{(k+1)/k}→0. (5%)
k=1
4. Find the taylor series expansion of f(x)=√(x+1) in powers of x
and give the radius of convergence. (15%)
5. Find the unit tangent, the principal normal, and write an equation in
x,y,z for the osculating plane at the point on the curve
→ → → →
r(t)=(sint-tcost)i+(cost+tsint)j +t^2/2 k at t=0. Parametrize the curve
by its arc length for t≧0. (20%)
→ → →
6. Suppose the curve r(t)=x(t)i+y(t)j is twice differentiable. If this curve
has nonzero tangent vector. Prove its curvature is
' " ' " ' '
│x(t)y(t)-y(t)x(t)│/([x(t)]^2+[y(t)^2])^3/2 .(10%)
7. Set f(x,y)=(x-y^4)/(x^3-y^4). Find the domain and range of f. Determine
whether or not f has a limit at (1,1). (10%)
8. Suppose that f is differentiable at (x ,y ) and ▽f(x ,y )≠0. Calculate
0 0 0 0
→ →
the rate change of f in the direction of f (x ,y )i-f (x ,y )j.
y 0 0 x 0 0
Give a geometric interpretation to your answer. (10%)
→ → → →
9. Set u(x,y,z)=u (x)i+u (x,y)j+u (x,y,z)k with x=x(t),y=y(t),z=z(t).
1 2 3
→
Derive a formula for du/dt. (10%)
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