**– $f(x,y) = |x| + |y|$**

**– $f(x,y) = |xy|$**

**– $f(x,y) = \frac{1}{1+x^2+y^2} $**

**– $f(x,y) = (x^2 – y^2)^2 $**

**– $f(x,y) =(x-y)^2$**

**– $f(x,y) = sin (|x| + |y|)$**

This question aims to find the** best graph match** for the given **functions** by using the concepts of **Calculus**.

This question uses the basic concepts of **Calculus** and **linear algebra** by **matching** the functions to the **best** contour graphs. **Contour graphs **simply **map** the two-dimension **input function** and **output functio**n of **one dimension**. The basic **figure** of the contour graph is shown below:

## Expert Answer

a)$f(x,y) = |x| + |y|$:

Suppose f(x,y) is equal to **Z**, then we have **Z equal to |x**| when the value of **y is zero** while **Z is equal to |y|** when the value of x is zero. So for this equation, the **best graph is labeled VI**.

b) $f(x,y) = |xy|$:

Suppose f(x,y) is equal to **Z**, then we have **Z** equal to **zero** when the value of **y** is **zero** while Z is equal to **zero** when the value of x is zero. So for this equation,** the best graph is labeled V**.

c) $f(x,y) = \frac{1}{1+x^2+y^2} $:

Suppose f(x,y) is **equal to Z**, so when the value of x is **zero**, we get

\[\frac{1}{1+y^2}\]

and when the value of y is **zero**, then we have:

\[\frac{1}{1+x^2}\]

When the value of **x** and **y** is very large, it will result in a zero value for** Z** so the best **match graph is I**.

d) $f(x,y) = (x^2 – y^2)^2 $:

Suppose f(x,y) is** equal to Z**, then the value of **x is zero**, we have:

\[Z=y^4\]

and when the value of **y** is **zero**, we have:

\[Z=x^4\]

and if **Z** is equal to** zero** then:

\[y=x\]

so the **best graph match is IV**.

e) $f(x,y) =(x-y)^2$:

Suppose f(x,y) is equal to Z, then the value of x is zero, we have:

\[Z=y^2\]

and when the value of **y is zero**, we have:

\[Z=x^2\]

and if Z is equal to zero then:

\[y=x\]

so the best graph match is II.

f) $f(x,y) = sin (|x| + |y|)$:

Suppose f(x,y) is equal to Z, then the value of x is zero, we have:

\[sin(|y|)\]

and when the value of y is zero, we have:

\[sin(|x|)\]

so the best graph match is III.

## Numeric Result

By assuming the values of $x$ and $y$, the given functions are matched in the best **contour graph.**

## Example

Draw the graph for function **$f(x,y) = cos(|x|+|y|)$.**

Suppose f(x,y) is** equal to Z**, then the value of **x is zero**, we have:

\[cos(|y|)\]

and when the value of **y is zero**, we have:

\[cos(|x|)\]

so the** best graph** for the **given function** is as follows:

*Images/Mathematical drawings are created with Geogebra.*