課程名稱︰微積分甲下
課程性質︰
課程教師︰吳德琪
開課系所︰電資學院
考試時間︰忘了...囧
試題 :
x^3-y^3 1
1. Let f(x,y)= ------- , g(x,y)= xy*sin(-------)
x^2+y^2 x^2+y^2
Then f,and g are not well-defined at (0,0)
●(10%) Give appropriate definition for f and g at (0,0),
such that f and g become continous functions at (0,0).
●(10%) Describe the diffentiability at (0,0) for f and g after
the above continuous extension.
2. Let f(x,y)=ln|xy|
●(20%) Find the tangent plane of f at (3,3).
●(20%) Estimate f(3.001,2.999). (ln3≒1.098)
3. Let f(x,y)=x^2 * y * exp(-x^2-y^2) (exp is exponential fuction)
g(u1,u2,t)= f( 1+t*u1 , 2+t*u2 )
Therefore, for any fixed (u1,u2), the set of points
(1+t*u1 , 2+t*u2 , g(u1,u2,t) )
is a curve on the surface z=f(x,y).
●(20%) Find (a,b) such that
δg δg
--- (a,b,0) = Maximum{ --- (u1,u2,0) | (u1)^2 + (u2)^2 =1}
δt δt
(δ is the sign for partial derivative)
●(20%) Find all (u1,u2) such that if g(u1,u2,t) has a local
extremum at t=s (s is a non-zero constant), then f(x,y)
also has a local maximum at ( 1+s*u1 , 2+s*u2 ).
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