Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 5

Domain: $x: (-\infty,\infty); y: (-\infty,\infty); z: (-\infty,\infty);$ Range: $f(x,y,z): (-\infty,\infty);$ Level surface: $f(x,y,z)=c$ (constant)

Work Step by Step

Function: $f(x,y,z)=x^{2}+y^{2}-z$ Domain: x, y, z can be any real numbers and (x,y,z) can be any point in the X-Y-Z system. Range: $f(x,y,z)$ can be any real number. Level surface: $f(x,y,z)=c$ (constant) $x^{2}+y^{2}-z=c$ $z=x^{2}+y^{2}-c$ represent a series of paraboloids. For example, when $c=-1$, $z=x^{2}+y^{2}+1$ can be plotted as shown in the figure.

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