Problem 23 - Calculus Early Transcendentals

Question

Use what you learned about surfaces in Section 1 to sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. $$p(x, y)=x^{2}-y^{2}.$$

Step-by-Step Solution

Answer

Answer: The given function represents a hyperbolic paraboloid. The domain is $\{(x, y) \mid x \in \mathbb{R}, \, y \in \mathbb{R}\}$, and the range is $\{z \mid z \in \mathbb{R}\}$.

Step 1: Rewrite the equation in another form

First, let's rewrite the given equation in terms of z by setting $$z = p(x,y) = x^2 - y^2$$. Now the equation looks like $$z = x^2 - y^2.$$

Step 2: Identify the surface

The given equation is a hyperbolic paraboloid. This can be seen by noticing the similarity between our equation and the general equation of a hyperbolic paraboloid, which is $$z = Ax^2 - By^2.$$ Here, A and B are both equal to 1.

Step 3: State the domain and range

The domain of a function is the set of all possible input values (x and y in this case) for which the function is defined. Since this is a hyperbolic paraboloid, there are no restrictions on the values that x and y can take. Therefore, the domain is: $$\text{Domain} = \{(x, y) \mid x \in \mathbb{R}, \, y \in \mathbb{R}\}.$$ The range of a function is the set of all possible output values (z in this case) that can be obtained for the given input values. For this hyperbolic paraboloid, the minimum value of z occurs when x and y are both 0, and z can otherwise take any positive or negative value. Therefore, the range is: $$\text{Range} = \{z \mid z \in \mathbb{R}\}.$$

Step 4: Sketch the graph

Now we will sketch the graph of the function. To do this, we can first sketch the cross sections of the hyperbolic paraboloid in the xz-plane and the yz-plane. In the xz-plane (when y = 0), the equation becomes: $$z = x^2.$$ This is a simple upward-facing parabola, symmetric about the x-axis. In the yz-plane (when x = 0), the equation becomes: $$z = -y^2.$$ This is a downward-facing parabola, symmetric about the y-axis. When combining these cross sections, we get a hyperbolic paraboloid with its vertex at the origin (0,0,0), with upward-opening parabolas along the x-axis, and downward-opening parabolas along the y-axis. Z increases as we move further away from the origin along the x-axis and decreases as we move further away from the origin along the y-axis.

Key Concepts

Hyperbolic Paraboloid

The hyperbolic paraboloid is a special type of surface in multivariable calculus. Think of it as a saddle-shaped structure that curves in two different directions. This surface is mathematically represented by the equation \(z = x^2 - y^2\), where the "positive" part \(x^2\) and the "negative" part \(-y^2\) give it its unique shape.

Some important features of a hyperbolic paraboloid include:

• Saddle Point: The point where the surface is neither rising nor falling, here, at the origin (0, 0, 0). This is where the surface has a horizontal tangent plane.
• Opposite Curvature: It curves upwards along one axis (\(x\)-axis) and downwards along the other (\(y\)-axis), making it a classic example of negative Gaussian curvature.

Understanding the shape and characteristics of the hyperbolic paraboloid is essential for visualizing and analyzing the problem.

Domain and Range

In calculus, the domain and range are crucial components of a function that tell us about where the function exists and what values it can output. The domain of a function of two variables, like \(p(x, y) = x^2 - y^2\), is the set of all possible pairs of \(x\) and \(y\) that can be plugged into the function.

For functions representing surfaces like hyperbolic paraboloids:

• The domain of \(p(x, y)\) is \(\{(x, y) \mid x \in \mathbb{R}, y \in \mathbb{R}\}\). This means \(x\) and \(y\) can be any real numbers.
• The range of \(p(x, y)\) is \(\{z \mid z \in \mathbb{R}\}\). Given the nature of the equation, \(z\) can take any real value, positive or negative.

By analyzing the domain and range, you can better understand the extent and bounds of your graph, in terms of both input and output values.

Graph Sketching

Graphing surfaces like the hyperbolic paraboloid involves understanding cross-sections and their appearances on different planes. By simplifying the function to two dimensions, we can better visualize the graph.

Here’s how you do it:

• In the \(xz\)-plane (where \(y = 0\)), the surface looks like a parabola \(z = x^2\), opening upwards.
• In the \(yz\)-plane (where \(x = 0\)), it becomes another parabola \(z = -y^2\), but this time opening downwards.

The final shape, when combining these cross sections, is like a saddle, with upward and downward parabolas hugging the axes. Practicing the steps of combining different planar views allows you to visualize the full graph that appears as a hyperbolic paraboloid.

Functions of Two Variables

Functions of two variables are an exciting area in multivariable calculus because they extend the concept of a function from a straight line to a surface. For example, \(p(x, y) = x^2 - y^2\) uses both \(x\) and \(y\) as input variables to produce an output \(z\).

Key aspects include:

• Input and Output Relationship: With each pair of inputs \((x, y)\), you get a single output \(z\), mapping a flat plane to an elevation.
• Real-World Applications: Such functions can model numerous real-world scenarios like temperature distributions, economic surfaces, or even optimizing certain constraints.

Understanding how two-variable functions operate aids in grasping more complex mathematical concepts and applications, making them a foundational element in advanced calculus studies.