Final answer:
The level surface of the function for the given point (1, 1, √2) is a sphere with a radius of 2. The equation of this sphere, centered at the origin, is x² + y² + z² = 4.
Step-by-step explanation:
The question asks to find an equation for the level surface of the function f(x, y, z) = √x² + y² + z² that passes through the point (1, 1, √2). The function f(x, y, z) describes the distance from the origin to the point (x, y, z) in three-dimensional space, which is also known as the Euclidean norm or the magnitude of a vector from the origin to (x, y, z). A level surface of this function represents a set of points that are all the same distance from the origin; in other words, a sphere centered at the origin.
To find the equation of the sphere that goes through the point (1, 1, √2), we first find the distance from the origin to that point using the function:
r = f(1, 1, √2) = √(1² + 1² + (√2)²) = √(1 + 1 + 2) = √4 = 2
Since the radius r is 2, the equation of the level surface (the sphere) is:
x² + y² + z² = r²
Substituting the radius we found:
x² + y² + z² = 2²
x² + y² + z² = 4
This is the desired equation for the level surface of the function at the radius determined by the given point.