Answer to Question #215073 in Calculus for Sarita bartwal

Question #215073
Find the domain 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
2021-07-08T13:58:29-0400

1.f(x,y,z)=zx2y2f(x,y,z)= \frac{z}{x^2-y^2}


dom(f)={(x,y,z)R3  x2y20}={(x,y,z)R3  (xy)(x+y)0}=R3{(x,x,z),(x,x,z))  x,zR}.dom(f)=\{(x,y,z)\in \R^3\ | \ x^2-y^2\ne 0\}=\{(x,y,z)\in \R^3\ | \ (x-y)(x+y)\ne 0\}= \R^3\setminus\{(x,x,z),(x,-x,z))\ |\ x,z\in\R\}.


2. f(x,y)=xsin(1x)+ysin(1y)f(x,y)= x \sin(\frac{1}{x})+ y \sin(\frac{1}{y})


dom(f)={(x,y)R2  x0,y0}=R2{(x,0),(0,y)  x,yR}.dom(f)=\{(x,y)\in\R^2\ |\ x\ne 0, y\ne 0\}=\R^2\setminus\{(x,0),(0,y)\ |\ x,y\in\R\}.


3. f(x,y,z)=14x2y2z2f(x,y,z)= \frac{1}{\sqrt{4-x^2-y^2-z^2}}


dom(f)={(x,y,z)R3  4x2y2z2>0}={(x,y,z)R3  x2+y2+z2<4}.dom(f)=\{(x,y,z)\in\R^3\ |\ 4-x^2-y^2-z^2>0\}=\{(x,y,z)\in\R^3\ |\ x^2+y^2+z^2<4\}.



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