Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch (as best you can) the graph of the monkey saddle \(z=x\left(x^{2}-3 y^{2}\right)\). Begin by noting where \(z=0\).

Step-by-Step Solution

Answer

The monkey saddle is zero on the lines \(x=0\), \(x=\sqrt{3}y\), and \(x=-\sqrt{3}y\).

Step 1: Understand the Equation

The equation of the monkey saddle is given by \(z = x(x^2 - 3y^2)\). We are tasked with sketching this surface, particularly noting the points where \(z = 0\). This surface is a 3D plot in the variables \(x\), \(y\), and \(z\).

Step 2: Determine Where \(z=0\)

To find where \(z=0\), set \(z = x(x^2 - 3y^2) = 0\). This equation will be zero if either \(x = 0\) or \(x^2 - 3y^2 = 0\).

Step 3: Analyze \(x=0\)

If \(x = 0\), then \(z\) will always be zero regardless of the value of \(y\). This means the entire \(y\)-axis is on the plane \(z=0\), and this line is a part of the monkey saddle surface.

Step 4: Solve \(x^2 - 3y^2 = 0\)

For \(x^2 - 3y^2 = 0\), rearrange to find \(x = \pm \sqrt{3}y\). This indicates two lines, \(x = \sqrt{3}y\) and \(x = -\sqrt{3}y\). Points on these lines will also have \(z=0\). These lines are also part of the surface on the plane \(z=0\).

Step 5: Sketch the Points Where \(z=0\)

Plot the lines \(x = 0\), \(x = \sqrt{3}y\), and \(x = -\sqrt{3}y\) in the \(xy\)-plane. All these lines are the intersections of the surface with the plane \(z=0\).

Step 6: Sketch the Overall Surface

Beyond the lines where \(z=0\), the monkey saddle surface will have elevations and depressions. The saddle's pattern involves positive and negative heights, forming a 'monkey saddle' shape. Sketch the overall 3D shape accounting for both zero and non-zero \(z\) values, noting symmetry and saddle behavior.

Key Concepts

3D Plotting: Visualizing the Monkey Saddle

One of the key tasks in this problem is creating a 3D plot of the monkey saddle surface described by the equation \(z = x(x^2 - 3y^2)\). 3D plotting refers to the process of graphing functions in three dimensions, allowing us to visualize complex surfaces. In this case, the monkey saddle surface has unique characteristics:

• The equation involves three dimensions: \(x\), \(y\), and \(z\). This means the graph will show how \(z\) changes depending on different values of \(x\) and \(y\).
• The surface is symmetrical around certain lines and points, key to understanding its behavior.

When plotting the monkey saddle, start by identifying regions where \(z = 0\). We found earlier that these include the entire \(y\)-axis and lines \(x = \pm \sqrt{3}y\). These features will guide the shape of our 3D graph.To further illustrate the surface, consider elevations and depressions across the plane. The challenge here is to accurately represent the 'saddle' nature, with its high and low points. Many software tools can assist in 3D plotting to visualize this effectively. Being familiar with such tools can be incredibly helpful in creating an accurate depiction.

Saddle Point: Understanding Surface Properties

The term "saddle point" is crucial in this context. A saddle point on a surface like the monkey saddle is a point that acts like a minimum in one direction and a maximum in another.In mathematical terms, you'll often find saddle points where the gradient is zero but are neither local maxima nor minima. For the monkey saddle, we can analyze the nature of the saddle point:

• The origin \((0,0,0)\) is a saddle point. Here, the slope changes direction, making it a point of interest in the surface sketch.
• In examining the monkey saddle, observe how movement along \(x\) or \(y\) affects the height \(z\). Along some paths, \(z\) will increase, while along others, it will decrease.

Visualizing this helps us understand why the surface is called a "saddle." It resembles a horse's saddle, with its characteristic ups and downs. Recognizing such points is essential when sketching the surface, contributing to a more accurate and dynamic understanding of the surface's geometry.

Surface Sketching: Creating a Visual Representation

Surface sketching is about capturing the essence of a 3D surface on paper or a screen. For the monkey saddle, it involves identifying its unique features and replicating them visually. Here are some simple steps:

• Start by plotting intersections with the plane \(z = 0\), such as the \(y\)-axis and the lines \(x = \pm \sqrt{3}y\). These lines serve as guides for your sketch.
• Next, try to imagine how the surface behaves beyond these lines. A surface like the monkey saddle will rise and fall, creating regions of elevation and depression. This is where the 'saddle' part of the name comes into play.
• Finally, incorporate symmetry. The surface displays a pattern not just along lines but around the point \( (0,0,0) \). Note how the surface appears similar on both sides of some axis, which helps in drawing a balanced sketch.

A successful surface sketch combines mathematical accuracy with intuitive visualization. Don't worry if it isn't perfect. The goal is to understand the primary shape and symmetry of the surface, which will deepen your grasp of three-dimensional graphing and problem-solving.