Problem 52 - University Calculus: Early Transcendentals

Question

Plot the surfaces in Exercises over the indicated domains. If you can, rotate the surface into different viewing positions. \(z=x^{2}+2 y^{2} \quad\) over a. \(-3 \leq x \leq 3,-3 \leq y \leq 3\) b. \(-1 \leq x \leq 1,-2 \leq y \leq 3\) c. \(-2 \leq x \leq 2,-2 \leq y \leq 2\) d. \(-2 \leq x \leq 2, \quad-1 \leq y \leq 1\)

Step-by-Step Solution

Answer

Plot the surface of \(z = x^2 + 2y^2\) for each specified domain using a graphing tool.

Step 1: Understand the Function

The function given is \(z = x^2 + 2y^2\). It represents a 3D surface. This function is a type of paraboloid, which is a 3D equivalent of a parabola.

Step 2: Define Domain for Part (a)

For part (a), the domain is defined as \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\). This will include all points \((x, y)\) within this range to plot the surface.

Step 3: Plot Surface for Part (a)

Set up a 3D plot within the specified domain for part (a). Use software or graphing tools to input the function \(z = x^2 + 2y^2\) and generate the surface plot.

Step 4: Define Domain for Part (b)

For part (b), the domain is \(-1 \leq x \leq 1\) and \(-2 \leq y \leq 3\). Use points in this range to plot part of the surface.

Step 5: Plot Surface for Part (b)

Input the specified domain for part (b) in your plotting tool alongside the function \(z = x^2 + 2y^2\) to visualize the surface over this restricted area.

Step 6: Define Domain for Part (c)

The domain for part (c) is \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\). Use these bounds to find the relevant section of the surface.

Step 7: Plot Surface for Part (c)

Generate the surface using \(z = x^2 + 2y^2\) for the domain defined in part (c). Examine how the surface appears within this smaller area.

Step 8: Define Domain for Part (d)

In part (d), the domain is given as \(-2 \leq x \leq 2\) and \(-1 \leq y \leq 1\). Use these intervals to explore a distinct portion of the surface.

Step 9: Plot Surface for Part (d)

Plot the surface over the domain specified in part (d) by inputting it into your graphing software along with the function \(z = x^2 + 2y^2\).

Key Concepts

Paraboloid

In 3D mathematics, a paraboloid is a surface that can be defined by a quadratic equation like the one given in this exercise: \(z = x^2 + 2y^2\). A parabola is a U-shaped curve, and when extended into three dimensions, it forms a paraboloid. This specific paraboloid is called an elliptic paraboloid, and it is shaped like an elongated dome that opens upwards.

This means for any fixed \(x\) and \(y\), the surface curves outward. An important characteristic of a paraboloid is its vertex, which is the point where the surface is smallest or largest. In this case, the vertex of the paraboloid is at the origin point \((0, 0, 0)\) when \(x\) and \(y\) are both zero. By understanding the geometry behind paraboloids, we can visualize how the surface behaves as \(x\) and \(y\) vary.

Domain in Multivariable Calculus

The concept of a domain in multivariable calculus is crucial for mapping out specific regions on a surface. In this exercise, the domain is defined by the boundaries for \(x\) and \(y\) values. For a thorough understanding, consider the domain as the "rectangle" or "region" in the xy-plane where you want to examine the behavior of the function \(z = x^2 + 2y^2\).

The domains provided, such as \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\), form a square space on the xy-plane where the surface will be plotted. Domains can be adjusted to zoom in or out on different parts of the 3D plot. These restrictions help examine specific portions of the surface, which can reveal different characteristics or patterns that might not be visible if the entire surface was shown at once.

Understanding and manipulating the domain is key in multivariable calculus because it allows us to focus on particular areas to study functions in a controlled manner.

Surface Visualization

Surface visualization involves creating a graphical representation of a mathematical function in three dimensions. By plotting a surface, like our elliptic paraboloid \(z = x^2 + 2y^2\), we can better understand the complex relationships between the variables \(x\), \(y\), and \(z\).

Using graphing software or programs, such as MATLAB or Python's Matplotlib, allows one to input the function and specify the domain to visualize the surface accurately. These tools create a plot that can often be rotated and adjusted for different perspectives. Viewing the surface from various angles exhibits its 3D structure more clearly. It can help identify features such as curvature, symmetry, and the impact of changes in the domain. For instance, from the top view, the elliptic nature of the paraboloid stands out, but tilting the view can illustrate the dome-shaped ascent of the surface.

• Interactive tools make it easier to explore and analyze the behavior of surfaces dynamically.
• Visualizing these surfaces enhances comprehension, bridging the gap between abstract math concepts and real-world applications.