Q. 14 - Calculus

Question

Explain why the function $z=g(x,y)=f\left(\sqrt{x^2+y^2}\right)$ is the equation of the surface obtained when the graph of $f$ is rotated about the $z$-axis. Sketch the surface obtained when your function from Exercise 13 is rotated about the $z$-axis.

Step-by-Step Solution

Answer

The function $z=g(x,y)=f\left(\sqrt{x^2+y^2}\right)$ is associated with two variable $x$ and $y$. When the function is rotated about $z$ - axis, a surface is generated in two variables $x$ and $y$.

Step 1: Recognize the composition

The function \(z = f(\sqrt{x^2+y^2})\) depends on \(x\) and \(y\) only through \(r = \sqrt{x^2+y^2}\), the distance from the origin in the \(xy\)-plane.

Step 2: Explain the consequence

Since \(z\) depends only on the distance \(r\) from the origin, the graph has rotational symmetry about the \(z\)-axis. The function has the same value at all points equidistant from the origin, making it a surface of revolution.