AMATHS一條f.4,考試啦,快快快,40分

The Q is solved as follows:

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2008-06-17 23:24:24 補充：
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1. Find dy/dx
d/dx (x^2+6xy-7y^2 )=d/dx(8)
2x+6[(y) dx/dx+(x) dy/dx]-7(2y)(dy/dx)=0
2x+6y+(6x)(dy/dx)-(14y) dy/dx=0
(x+3y)+(3x-7y)(dy/dx)=0
dy/dx=-(x+3y)/(3x-7y)

2. Find d^2y/dx^2
d/dx dy/dx=d/dx (-(x+3y)/(3x-7y))
(d^2 y)/(dx^2 )=-((3x-7y)d/dx (x+3y)-(x+3y)d/dx(3x-7y))/〖(3x-7y)〗^2
(d^2 y)/(dx^2 )=-((3x-7y)(1+3 dy/dx)-(x+3y)(3-7 dy/dx))/〖(3x-7y)〗^2
(d^2 y)/(dx^2 )=-((3x-7y-3x-9y)+(9x-21y+7x+21y)dy/dx)/〖(3x-7y)〗^2
(d^2 y)/(dx^2 )=-((-16y)+(16x)dy/dx)/〖(3x-7y)〗^2
(d^2 y)/(dx^2 )=-(-16y+16x(-(x+3y)/(3x-7y)))/〖(3x-7y)〗^2
(d^2 y)/(dx^2 )=-(16)(-y(3x-7y)-x(x+3y))/〖(3x-7y)〗^3
(d^2 y)/(dx^2 )=(16(x^2+6xy-7y^2))/〖(3x-7y)〗^3
(d^2 y)/(dx^2 )=(16(8))/〖(3x-7y)〗^3 (Given)
(d^2 y)/(dx^2 )=128/〖(3x-7y)〗^3

Question: x^2+6xy-7y^2=8

1)Differentiate both sides with respect to x, we have

2x+6x(dy/dx)+6y-7(2y)(dy/dx)=0
2(x+3y)=14y(dy/dx)-6x(dy/dx)
x+3y=(7y-3x)(dy/dx)
dy/dx=(x+3y)/(7y-3x)

2)Differentiate the result in part(1) with respect to x, we have
d^2y/dx^2={(7y-3x)[1+3(dy/dx)]-(x+3y)[7(dy/dx)-3]}/(7y-3x)^2
={(7y-3x)[1+3(x+3y)/(7y-3x)]-(x+3y)[7(x+3y)/(7y-3x)-3]}/(7y-3x)^2
=[(7y-3x)+3(x+3y)-7(x+3y)^2/(7y-3x)+3(x+3y)]/(7y-3x)^2
=[(7y-3x)+6(x+3y)-7(x+3y)^2/(7y-3x)]/(7y-3x)^2
=[(3x+25y)(7y-3x)-7(x+3y)^2]/(7y-3x)^3
=(175y^2-54xy-9x^2-7x^2-42xy-63y^2)/(7y-3x)^3
=(-16x^2-96xy+112y^2)/(7y-3x)^3
=-16(x^2+6xy-7y^2)/(7y-3x)^3
=-16(8)/(7y-3x)^3
=128/(3x-7y)^3