Answer to Question #219211 in Calculus for Sarita bartwal

Question #219211
Find the range of 1.f(x,y,z)= z/(x^2-y^2)
2. f(x,y)= x sin(1/x)+ y sin(1/y)
3. f(x,y,z)= 1/(√ (4-x^2-y^2-z^2)
1
Expert's answer
2021-07-27T09:17:10-0400

Answer:-


1.


x2y20x^2-y^2\not=0

If z0,x2>y2,z\geq 0, x^2>y^2, then f(x,y,z)0.f(x, y,z)\geq0.

If z0,x2<y2,z\geq 0, x^2<y^2, then f(x,y,z)0.f(x, y,z)\leq0.

If z0,x2>y2,z\leq 0, x^2>y^2, then f(x,y,z)0.f(x, y,z)\leq0.

If z0,x2<y2,z\leq 0, x^2<y^2, then f(x,y,z)0.f(x, y,z)\geq0.

Range: (,)(-\infin, \infin)


2. Let u(x)=xsin(1x)u(x)=x\sin(\dfrac{1}{x})

If x±,x\to\pm \infin, then u(x)1u(x)\to 1^{-}



u=sin(1x)1xcos(1x)u'=\sin(\dfrac{1}{x})-\dfrac{1}{x}\cos(\dfrac{1}{x})u=0=>sin(1x)1xcos(1x)=0u'=0=>\sin(\dfrac{1}{x})-\dfrac{1}{x}\cos(\dfrac{1}{x})=0x1=0.22255,x2=0.22255x_1=-0.22255, x_2=0.22255u(0.22255)=u(0.22255)0.21723u(-0.22255)=u(0.22255)\approx-0.217230.21723u<1,x0-0.21723\leq u<1, x\not=00.217230.217230.4345-0.21723-0.21723\approx-0.4345



Range: [0.4345,2)[-0.4345, 2)


3.


4x2y2z2>04-x^2-y^2-z^2>00x2+y2+z2<40\leq x^2+y^2+z^2<4

Then



0<4x2y2z220<\sqrt{4-x^2-y^2-z^2}\leq2

Range: [12,)\bigg[\dfrac{1}{2}, \infin\bigg)




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