Question

Choose a phrase that describes the level surfaces of each of the following functions as the function value increases:

(a) f(x,y,z)=√9−x² −y²
(b) f(x,y,z)=√x²+y²+z²
(c) f(x,y,z)=1/(x²+y²+z²)
(d) f(x,y,z)=5+y²+z²

Answer 1

The level surfaces of the given functions change as the function value increases. For functions (a) and (b), the level surfaces are concentric spheres centered at the origin, while for function (c) they are also concentric spheres but move closer to the origin. Function (d) has level surfaces in the form of parallel planes perpendicular to the x-axis.

Step-by-step explanation:

The level surfaces of a function are the surfaces in three-dimensional space where the function takes on a constant value. To describe the level surfaces as the function value increases:

(a) For the function f(x, y, z) = √9 - x² - y², as the function value increases, the level surfaces move away from the origin and become more spread out. The level surfaces are concentric spheres centered at the origin, with radii ranging from 0 to 3.

(b) For the function f(x, y, z) = √x² + y² + z², as the function value increases, the level surfaces move away from the origin and become more spread out. The level surfaces are concentric spheres centered at the origin, with radii increasing.

(c) For the function f(x, y, z) = 1/(x² + y² + z²), as the function value increases, the level surfaces move closer to the origin and become more concentrated. The level surfaces are concentric spheres centered at the origin, with radii ranging from 0 to infinity.

(d) For the function f(x, y, z) = 5 + y² + z², as the function value increases, the level surfaces move away from the origin and become more spread out. The level surfaces are a stack of parallel planes, perpendicular to the x-axis, with spacing increasing as the function value increases.