Problem 26 - 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. $$H(x, y)=\sqrt{x^{2}+y^{2}}.$$

Step-by-Step Solution

Answer

Answer: The function $$H(x, y)=\sqrt{x^{2}+y^{2}}$$ represents a conical surface centered at the origin. The domain of the function is $$D = \{(x, y) \mid x, y \in \mathbb{R}\}$$, and the range is $$R = \{H(x, y) \mid H(x, y) \geq 0\}$$.

Step 1: Analyze the function

The given function is $$H(x, y)=\sqrt{x^{2}+y^{2}}.$$ We can see that the function is taking the square root of the sum of squares of \(x\) and \(y\). The function looks like it might represent the distance between a point \((x, y)\) on the plane and the origin.

Step 2: Identify the surface type

Since the function takes the form of square root of the sum of squares, it represents the distance between a point \((x, y)\) on the plane and the origin. This means that the function describes a conical surface centered at the origin. As we move further from the origin, the cone grows in height.

Step 3: Sketch the graph

To sketch the graph, we can imagine the cone starting at the origin and growing outwards along the positive and negative axes for both \(x\) and \(y\). The graph will be symmetrical with respect to the \(x\) and \(y\) axes, and will have a continuous surface.

Step 4: Determine the domain

The domain of the function consists of all possible inputs for the function. In this case, \(x\) and \(y\) can be any real numbers, as the sum of their squares will always produce a non-negative value. Therefore, the domain of the function is $$D = \{(x, y) \mid x, y \in \mathbb{R}\}.$$

Step 5: Determine the range

The range of the function consists of all possible outputs. In this case, since the function represents the distance between a point \((x, y)\) in the plane and the origin, it can never be negative. Also, the distance can be made arbitrarily large by choosing larger and larger values of \(x\) and \(y\). Therefore, the range of the function is $$R = \{H(x, y) \mid H(x, y) \geq 0\}.$$

Key Concepts

Graph of Functions

A graph or plot of a function in multivariable calculus visually represents the relationship between the input variables and the output of a function. For the function \(H(x, y) = \sqrt{x^2 + y^2}\), the graph depicts all the points \((x, y, H(x, y))\) in space.
When sketching the graph of the function \(H(x, y)\), envision a three-dimensional space where each \(x\) and \(y\) corresponds to a certain height \(z = H(x, y)\).
The surface created by this function looks like a cone, with its tip or vertex at the origin (0,0) on the xy-plane. As \(x\) and \(y\) increase, the height \(z\) rises, forming a circular cross-section that grows wider.
Key characteristics of the graph include:

• Symmetry about the z-axis: The function is symmetrical around the origin because \(H(x, y)\) depends only on the distance from the origin.
• Continuous surface: As \(x\) and \(y\) change, there are no breaks or gaps in the surface.

Understanding these elements helps in comprehending how the function behaves and interacts in a three-dimensional space.

Surface Identification

Surface identification in multivariable calculus involves determining the type of surface represented by a function. For \(H(x, y) = \sqrt{x^2 + y^2}\), the key is to note that the function's structure—for this example, it measures the Euclidean distance from the origin on the plane.
Such a function corresponds to a circular cone centered at the origin.
The surface is conical because:

• For each fixed \(z = c\), where \(c\) is a constant, the equation \(x^2 + y^2 = c^2\) represents a circle in the xy-plane.
• As \(c\) (or \(H(x, y)\)) increases, the radius of these circles grows, forming a cone that extends infinitely upwards.

The significance of this surface lies in its symmetry and shape, which are vital in many applications like physics and engineering, where such distances are used to model real-world phenomena.

Domain and Range

Determining the domain and range of a multivariable function involves identifying all possible inputs and outputs, respectively. For the function \(H(x, y) = \sqrt{x^2 + y^2}\), understanding these sets can enhance comprehension of how the function operates.

**Domain**:
The domain of \(H(x, y)\) includes all combinations of \(x\) and \(y\) that yield a defined value from the function. Here, since \(x^2 + y^2\) is always non-negative for any real numbers \(x\) and \(y\), the domain is the entire xy-plane. Thus,
\[ D = \{(x, y) \mid x, y \in \mathbb{R} \} \]
**Range**:
The range includes all possible values that \(H(x, y)\) can take. Since the function outputs the radial distance, it can yield any non-negative real number. Hence, the range is given by:
\[ R = \{H(x, y) \mid H(x, y) \geq 0 \} \]
Recognizing the domain and range helps in visualizing the possible extents of a function and understanding limitations or opportunities when analyzing multivariable systems.