課程名稱︰微積分乙下
課程性質︰必修
課程教師︰張志中
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014.05.06
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. (20%)
Find and classify the critical points of h(x,y)=3x^2y+y^3+3x^2+3y^2.
2. (30%)
Let F(x,y,z)=x^2y+y^2z+z^2x and S_1 be the level surface F(x,y,z)=5.
Let G(x,y,z)=x^3y^3+y^2z^2+zx and S_2 be the level surface G(x,y,z)=5.
(a)
Find the plane tangent to S_1 at (1,-1,2).
(b)
Let C be the curve of the intersection of two surfaces S_1 and S_2. Find the
line tangent to C at (1,-1,2).
(c)
Let z=z(x,y) be the function implicitly defined by F(x,y,z)=5 around
(x,y,z)=(1,-1,2). Find the direction(s) at the point (x,y)=(1,-1) along which
z(x,y) decreases most rapidly. That is, find unit vector(s) u 屬於 R^2 such
that the directional derivtive (Du z)(1,-1) attains its minimum.
3. (20%)
Let u=u(t), v=v(t) be the differentiable functions solved from (defined by)
the equations
2u^2+v=e^(tv+2)
u^3v^2=e^(tu-2)
around t=2 and (u,v)=(1,-1).
(a)
Evaluate u'(2) and v'(2).
(b)
Evaluate H'(2), where H(t)=v(t)/u(t).
4. (30%)
Find the extreme values of f(x,y)=x^2y+2xy+xy^2 on x^2+xy+y^2+2x+2y+1=0.