Plotting a 3D Function Surface

Imagine you have a function $z = x^2 + y^2$. How would you begin to plot this function in a 3-dimensional space?

Plotting a function like $z = x^2 + y^2$ in three-dimensional space involves understanding how functions can represent surfaces. In this context, the given function illustrates a simple surface known as a paraboloid. Conceptually, each combination of x and y in the domain corresponds to a z value, creating a surface above the xy-plane. This exercise is foundational for exploring more complex surfaces and understanding the geometric representation of multivariable functions.

To effectively visualize this function, we begin with a grid of x and y values, typically covering a symmetric range around zero. The squared terms ensure a positive z value, showing that the function is always above or on the xy-plane, and the resulting graph is circularly symmetric around the z-axis. This symmetry is important as it simplifies the plotting process and can help in understanding the nature of the function's surface.

This problem also introduces key concepts such as level curves, which are cross sections of the surface at constant z values. These curves aid in visualizing the function's surface from a top-down view by representing contours, much like elevation lines on a map. Understanding level curves and their geometric significance is crucial for grasping more advanced topics in multivariable calculus, such as gradients and surface integrals.

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