課程名稱︰微積分甲下
課程性質︰大一共同必修
課程教師︰薛克民
開課學院：生農學院,理學院
開課系所︰工管科管組,生機系,生工系,地質系
考試日期（年月日）︰98/4/13
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(a) Consider a logistic population model of the form
dp p
── = 2p(1-─ )
dt 5
Find the solution p(t) with the initial condition p(0)=1,and then compute
lim p(t)
t→∞
(10 points)
(b)Find the solution y(x) that satisfies the following initial value
problem:
dy
x── -y=xlnx y(1)=2.
dx
(10 points)
2.(a) Find the area of the region that lies inside the curve r=2+cos2θ
but outside the curve r=2+sinθ ,0≦θ≦2π as drawn below
(10 points)
→
(b)Consider the curve r (t)=( 2t^3/2 , cos2t , sin2t) which is written in a
→
vector function form,t屬於R,Find the unit tangent vector T (t) for the curve
→ →
r (t) at t=0 , and the length of the curve r (t) , 0≦t≦1.
(10 points)
3.(a) Determine whether the series
∞ n+2
Σ ──
n=1 n+1
is convergent or divergent.
(8 points)
(b)Determine whether the series
∞ (-1)^n
Σ ───
n=1 n^1/3
is conditionally convergent,absolutely convergent, or divergent.
(7 points)
4.(a)Find the Taylor series of f(x) =sinx at a=π/6 (write down the general
term)
(10 points)
(b)Find the Maclaurin series for f(z)=1/(16-x)^-1/4 (write down at least 5
nonzero terms) and find its radius of convergence.
(10 points)
5.Find the equation of the tangent plane to z=4x^2-y^2+2y at (-1.2.4)
(10 points)
6.Consider z(z,y)=e^xcosy with x=st y=(s^2+t^2)^1/2 . Find
(以下以p 表示為偏微分記號)
2
pz p z
(a) ── (5 points) (b) ─── (10 points)
ps ps pt
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