To accomplish this in Mathematica, you can use the following steps to create the contour and surface plots as specified:

Part (i): Contour Plot of $g(x, y) = 3x \cos(x + y^2)$

1. Define the function $g(x, y) = 3x \cos(x + y^2)$.
2. Use ContourPlot with the specified range for $x$ and $y$.
3. Set the Contours option to 16 to show 16 different contour lines.
4. Add a legend to indicate the value of each contour.

Here's the Mathematica code for the contour plot:

mathematicaCopy code
g[x_, y_] := 3 x Cos[x + y^2]
ContourPlot[g[x, y], {x, -4.5, 4.5}, {y, -2.5, 2.5}, 
 Contours -> 16, ColorFunction -> "Rainbow", 
 PlotLegends -> Automatic, 
 ContourLabels -> True, FrameLabel -> {"x", "y"}, 
 PlotLabel -> "Contour Plot of g(x, y)"]

• ColorFunction -> "Rainbow" will apply a color gradient to enhance visualization.
• PlotLegends -> Automatic will add a legend that shows the contour values.
• ContourLabels -> True will display the values of the contours directly on the plot.

Part (ii): Surface Plot of $z = g(x, y)$

1. Use Plot3D to create a surface plot for $z = g(x, y)$.
2. Set the viewing range for $x$ and $y$ as specified.
3. Use the rotation tool in Mathematica to adjust the orientation for clarity.

Here’s the Mathematica code for the surface plot:

mathematicaCopy code
Plot3D[g[x, y], {x, -4.5, 4.5}, {y, -2.5, 2.5}, 
 PlotRange -> All, ColorFunction -> "Rainbow", 
 Mesh -> None, Boxed -> False, 
 AxesLabel -> {"x", "y", "z"}, 
 PlotLabel -> "Surface Plot of g(x, y)"]

• PlotRange -> All ensures that the full range of $z$-values is shown.
• ColorFunction -> "Rainbow" provides a color gradient across the surface.
• Mesh -> None removes the mesh lines for a cleaner look.
• You can use the interactive rotation tool in Mathematica to orient the plot to display the surface features as desired.

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Let me know if you'd like any further details on how these plots work or if you have additional questions. Here are some related questions for exploration:

1. How does changing the number of contours affect the readability of the contour plot?
2. What insights can be drawn by observing both the contour plot and the surface plot?
3. How does the choice of ColorFunction affect the clarity of these plots?
4. What happens if we extend the range of $x$ and $y$ for this function?
5. How can we use the Manipulate function in Mathematica to dynamically explore changes in parameters?

Tip: Experiment with different ColorFunction options in Mathematica to find one that best highlights features in your plot.