Question

1. match each of graphs (a) and (b) in figure 22 with one of the following functions: (i) f(x,y)=(cos x)(cos y) (ii) g(x,y)=cos(x^2 + y^2)

Explanation:

Step1: Analyze function (i) $f(x,y)=(\cos x)(\cos y)$

This function is a product of two - one - variable cosine functions. When $x = k\pi$ ($k\in\mathbb{Z}$), $f(x,y)=0$ for all $y$, and when $y = m\pi$ ($m\in\mathbb{Z}$), $f(x,y)=0$ for all $x$. It has a grid - like pattern of zeros in the $xy$ - plane. The graph will have a series of ridges and valleys that are aligned with the $x$ and $y$ axes.

Step2: Analyze function (ii) $g(x,y)=\cos(x^{2}+y^{2})$

This is a function of the form $\cos(r^{2})$ where $r^{2}=x^{2}+y^{2}$. It is a circularly symmetric function. The level - curves of $g(x,y)$ are circles centered at the origin. As $r^{2}=x^{2}+y^{2}$ increases, the function $\cos(x^{2}+y^{2})$ oscillates.

Step3: Match the functions with the graphs

Function (i) $f(x,y)=(\cos x)(\cos y)$ corresponds to graph (A) because of its grid - like pattern of zeros. Function (ii) $g(x,y)=\cos(x^{2}+y^{2})$ corresponds to graph (B) due to its circular symmetry.

Answer:

(i) matches (A), (ii) matches (B)