Problem 7 - Calculus Early Transcendentals

Question

Describe in words the level curves of the paraboloid \(z=x^{2}+y^{2}\).

Step-by-Step Solution

Answer

Based on the solution above, the level curves of the paraboloid \(z=x^{2}+y^{2}\) are circles centered at the origin with a radius determined by the value of z. As z increases, the circles have larger radii, and when z = 0, the level curve is just the origin point.

Step 1: Understand the given function

The given function is a paraboloid with the equation \(z=x^{2}+y^{2}\). It represents a 3D surface in which the value of z depends on x and y.

Step 2: Set up the level curve equation

A level curve for a function is a curve in which the function has a certain constant value. In our case, we want to find the level curves of the paraboloid, so we will set the equation equal to a constant: $$x^{2}+y^{2}=c$$ where c is a constant representing the value of z for the level curve.

Step 3: Evaluate level curves for different values of c

We will now examine the equation for different values of c, which represents different values of z for the level curves: 1. For c = 0: $$x^{2}+y^{2}=0$$ This is a degenerate case, as x and y must both be 0. Therefore, when z = 0, the level curve is just a single point at the origin (0, 0). 2. For c > 0: $$x^{2}+y^{2}=c$$ This equation represents a circle centered at the origin with a radius of \(\sqrt{c}\). As c increases, the circles will have larger radii.

Step 4: Describe the level curves

The level curves of the paraboloid \(z=x^{2}+y^{2}\) are circles centered at the origin (0, 0). The radius of each circle is determined by the value of z, resulting in a family of nested circles for different values of z. As z increases, the radius of the circles also increases. When z = 0, the level curve is just a single point at the origin.