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A contour map is a two-dimensional representation of a three-dimensional surface, showing lines of constant value, known as contour lines. In the context of our function \( f(x, y) = \sqrt{36 - 9x^2 - 4y^2} \), these contours are actually ellipses. By setting \( f(x, y) = c \), where \( c \) is a constant, we derive an equation in the form \( 9x^2 + 4y^2 = 36 - c^2 \). This kind of equation is fundamental to creating a contour map.

To sketch the contour map, you choose values of \( c \) and plot the resulting ellipses on the xy-plane. These ellipses will have their centers at the origin, and their shapes can change based on the value of \( c \). Each ellipse represents a cross-section of the original 3D surface at the height \( z = c \).

• Choose different values of \( c \) such as 0, 2, and 3 to get different ellipses.
• Draw these ellipses on the xy-plane.
• Observe how these ellipses shrink in size as \( c \) increases, because the function only considers the upper part of the ellipsoid.

When sketching a 3D graph of the function \( f(x, y) = \sqrt{36 - 9x^2 - 4y^2} \), you're visualizing a surface in three-dimensional space. This surface resembles the top half of an ellipsoid. The square root in the function limits the graph to non-negative values, creating a dome shape.

To graph it, consider these steps:

• Realize that the graph is symmetrical around the z-axis, originating from the xy-plane, giving it a dome-like appearance.
• Calculate the highest point of the graph by setting \( x = 0 \) and \( y = 0 \), which gives \( z = \sqrt{36} = 6 \).
• The surface diminishes as \( x \) and \( y \) move away from the center, eventually reaching a baseline of 0 when \( 36 - 9x^2 - 4y^2 = 0 \).

This process allows you to fully appreciate how each level contour translates into parts of the 3D structure.

An ellipsoid is a three-dimensional shape characterized by its elongated spherical surface. The function \( f(x, y) = \sqrt{36 - 9x^2 - 4y^2} \) describes the top half of an ellipsoid, which can be understood by looking at a rearrangement. The expression within the square root, \( 36 - 9x^2 - 4y^2 \), represents an ellipsoid centered at the origin when equated to z-values.

In a full ellipsoid equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), each division term represents the squares of axes lengths. Here, however, the equation devotes to the half, i.e., the top portion, since the square root function only covers positive z-values.

• The semi-principal axes align with the x and y axes where \( ax = \sqrt{4} \approx 2 \) and \( by = \sqrt{9} \approx 3 \).
• A z-axis line at the tallest point remains \( c = \sqrt{36} = 6 \).
• The symmetry along the xy-plane reflects a contrast between higher points in z and their reflections below, removed by the square root constraint.

Understanding this concept aids in grasping the inherent structure typified by an ellipsoid, crucial for visualizations in multivariable calculus.

An ellipse is a flat, 2D shape resembling a stretched circle. In the context of this exercise, each level curve of the contour map forms an ellipse. These ellipses arise when you fix a specific value for the constant \( c \) in the function equation \( 9x^2 + 4y^2 = 36 - c^2 \).

An ellipse's key features include:

• Centers: These are located at the origin \((0, 0)\) for this problem's contour map.
• Semi-major and semi-minor axes: For our elliptic contours in the xy-plane, determine axis length by rewriting \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) and scaling it negatively as \( c^2 \) increases.
• Eccentricity: A measure of an ellipse's deviation from a perfect circle, influenced by the ratios of the axes lengths drawn in varying levels for diverse \( c \).

Understanding ellipses helps understand sectioning of more complex shapes like ellipsoids and enhances interpreting multivariable calculus problems efficiently.