第1个回答 2011-12-31
1
y=kx
kx=x^2-x+2
x^2-(1+k)x+2=0
(1+k)^2-4*2=0
(1+k)^2=8
y=(-1+2√2)x 和y=(-1-2√2)x
y=-1+2√2x和y=x^2-x+2 相切于 A(√2,3)
y=-1-2√2x和y=x^2-x+2 相切于B(-√2,-5)
AB直线 y-3=[(-5-3)/2√2](x+√2)=-2√2x-4
y=-2√2x-1
y=-1+2√2x和y=-1-2√2x 交于P(0,-1)
Sabo=∫[-√2,0] [(x^2-x+2)-(-1-2√2x)] dx+∫[0,√2](x^2-x+2)-(-1+2√2x)]dx
=∫[-√2,0](x+√2)^2dx+∫[0,√2](x-√2)^2dx
=(1/3)2√2+(1/3)2√2=4√2/3
2
y^2=2x 切线斜率为y' 法线斜率为-1/y'=-x'
x'=y x'(0.5,1)=1
y-1= -(x-0.5)
y=-x+1.5
y^2=2x
y^2=2*(1.5-y)=3-2y
y^2+2y-3=0
y1=1,y2=-3
x1=1/2 x2=9/2
S=∫[-3,1] [(1.5-y)-y^2/2]dy=(1/2)∫[-3,1](-y^2-2y+3)dy
=(1/2)[(-1/3-1+3)-(-27/3-9+(-9)]
=(1/2)(18+32/3)
=9+16/3
第2个回答 2011-12-31
(1)
y’=2x-1
y=kx
kx=x^2-x+2
2x-1=k x=(k+1)/2
k(k+1)/2=((k+1)/2)^2-(k+1)/2+2
2k^2+2k=k^2+2k+1-2k-2+2
K^2+2k-1=0
K=2^0.5-1 k=-2^0.5-1
x1=2^0.5/2 x2=-2^0.5/2
y=(2^0.5-1)x y=(-2^0.5-1)x
∫(0,-2^0.5/2)(x^2-x+2-(-2^0.5-1)x)dx+∫(2^0.5/2,0) (x^2-x+2-(2^0.5-1)x)dx=-((-2^0.5/2)^3/3+2^0.5 (-2^0.5/2)^2/2+2(-2^0.5/2))+ (2^0.5/2)^3/3-2^0.5 (2^0.5/2)^2/2+2(2^0.5/2)
=(2^0.5/2)^3/3-2^0.5 (2^0.5/2)^2/2+2(2^0.5/2)+ (2^0.5/2)^3/3-2^0.5 (2^0.5/2)^2/2+2(2^0.5/2)
=2^0.5/12-2^0.5/4+2^0.5
=1.17851
(2)
y^2=2x
x=y^2/2
x’=y
y=1 x’=1
y-1=-(x-0.5) x=-y+1.5
x=y^2/2 x=-y+1.5
y^2/2=-y+1.5
y=1 y=2
∫(2,1)( -y+1.5- y^2/2)dy
=-y^2+1.5y-y^3/6|(2,1)
=-4+3-8/6+1-1.5+1/6
=2.6667