課程名稱︰微積分乙下
課程性質︰系必修
課程教師︰王振男
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2013/4/16
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題：
1.（a）（5％）Find dy/dx if cos(x^2＋y^2)＝sin(x^2－y^2).
（b）（5％）In what direction does f(x,y)＝e^xcosy increase most rapidly at
(0,π/2)？ ┌ ┐
｜x^2－xy｜
2.（10％）Let the vector-valued function f(x,y)＝｜3y^2－1｜.
└ ┘
Using the linear approximation to approximate f(1.1,1.9).
3.（10％）Solve the initial value problem：y'＝(1/2)y^2－2y with y(0)＝－3.
4.Denote the size of a population at time t by Ｎ(t), and assume that
dＮ Ｎ
───＝2Ｎ(Ｎ－10)(1－──) for t≧0.
dt 100
（a）（5％）Find all equilibria.
（b）（5％）Determine the stability of the equilibria you found in（a）.
5.Consider the initial value problem：┌ y'＝(y＋1)√(y^2＋2y)
└ y(0)＝0
（a）（10％）Solve (1).
（b）（10％）Does (1) have only one solution？If not, can all solution of (1)
be defined for any t＞0？
6.（15％）Use the method of Lagrange multiplier to find the point on the plane
2x＋y＋2z＝1 which is nearest to the origin.
7.（10％）Find the tangent line of the curve formed by the intersection of
x^2＋2(x^2)(y^2)＋(y^3)z＋e^z－2＝0 and z(e^xy)＋cos[y^3＋(x^2)(z^2)]－1＝0
at the point (1,0,0).
8.（15％）Find the maximum and minimum values of f(x,y)＝2x^2－y^2 over the
region {(x,y)：x^2＋2(y－1)^2≦4}.
參考答案：
xsin(x^2＋y^2)＋xcos(x^2－y^2)
1.－ ────────────────
ysin(x^2＋y^2)－ycos(x^2－y^2)
┌ ┐
2.│－0.9│
│ 9.8│
└ ┘
4
3. ────────
1－(7/3)(e^2x)
4.（a）Ｎ＝0,10,100；（b）Ｎ＝0,100 →stable、Ｎ＝10 →unstable
5.（a）y＝sec(t＋2aπ)－1（a∈Z）
（b）not；not.
6.(2/9,1/9,2/9)
7.x＝1,z＝0,y∈R. 7 4
8.maximum＝36/5 at points (±─√2,─)；minimum＝－3－2√2 at point (0,1+√2)
5 5
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