Use a graphing utility to sketch graphs of z=f(x, y) from...

Use a graphing utility to sketch graphs of $z=f(x, y)$ from two different viewpoints, showing different features of the graphs.
(a) $f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$
(b) $f(x, y)=2 x \sin x y \ln y$

Key Concepts

Composite Function Analysis

When a function is built by combining multiple elementary functions—such as polynomials, trigonometric functions, and logarithms—each part contributes its unique characteristics. Analyzing the composition and interaction of these components is essential for understanding the overall behavior of the graph.

Behavior of Logarithmic Functions

Logarithmic functions have well-known properties, including restrictions on their inputs and characteristic asymptotic behavior. Recognizing these properties in multivariable contexts aids in understanding where the function is defined and how it behaves near its boundaries.

Symmetry and Radial Structures

In cases where the function depends on expressions like the sum of squares of the variables, the function often exhibits symmetry, typically around the origin. Recognizing such symmetries can simplify both the analysis and the visualization, as circular or radial patterns might emerge.

Domain Analysis

Many multivariable functions have restrictions on the domain because certain operations, like taking logarithms, require the input to be in a specific range (for example, the input must be positive). Understanding and determining the domain is crucial, as it influences the behavior and the regions where the graph exists.

3D Surface Visualization

Graphing a function of two variables in three dimensions helps in visualizing its overall structure, such as peaks, valleys, and asymptotic behavior. Different viewpoints or rotations can reveal hidden features of the surface that may not be obvious from a single perspective.

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