Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Draw the graph and the corresponding contour plot. \(f(x, y)=\left(2 x-y^{2}\right) \exp \left(-x^{2}-y^{2}\right) ;-2 \leq x \leq 2\) \(-2 \leq y \leq 2\)

Step-by-Step Solution

Answer

Graph the function using a 3D plot and its level curves using a contour plot over the range \(-2 \leq x, y \leq 2\).

Step 1: Understand the Function

The function given is a product of two components: a quadratic polynomial, \(2x - y^2\), and an exponential decay term, \(\exp(-x^2 - y^2)\). The exponential term will ensure that the function smoothly decreases to zero as the values of \(x\) and \(y\) increase or decrease from the origin. This means we'll have a bell-shaped surface centered around the origin.

Step 2: Set Up the Grid for Plotting

For graphing purposes, we need a grid that spans the domains \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\). We'll use a meshgrid which creates two matrices, one for the \(x\) values and another for \(y\) values, that will be used in calculating \(f(x, y)\) for plotting.

Step 3: Calculate Function Values

With the grid set up, calculate the values of \(f(x, y)\) at each point in the grid. This involves computing both the quadratic polynomial \(2x - y^2\) and the exponential term \(\exp(-x^2 - y^2)\), then multiplying these components together for each combination of \(x\) and \(y\).

Step 4: Plot the 3D Surface Graph

Using a plotting library like Matplotlib in Python, plot the computed function values over the range. The function should appear as a 3D bell-shaped surface with some asymmetry due to the \(2x-y^2\) term causing a distortion.

Step 5: Plot the Contour Graph

Create a contour plot that represents the 'level curves' of our function. These curves represent points where the function has the same value. On Matplotlib, this plot can be created on the same \(x,y\) range grid used before and will visually show the topological features of the surface.

Key Concepts

Contour Plot

A contour plot is a way to show a 3D surface on a 2D plane using curves. It’s like a topographic map, where each line, or contour, represents points of the same value. When working with multivariable functions like our function, \( f(x, y) = (2x - y^2)\exp(-x^2 - y^2) \), contour plots can help visualize how the function behaves over a given domain.
Contour plots are useful because:

• They provide a clear, flat representation of peaks and valleys.
• They help identify areas of constant function value, which are helpful in optimization problems.

In our exercise, you would see lines indicating the level values across the grid from \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\), clearly showing how the values change across the plot.

3D Surface Graph

A 3D surface graph provides a visual landscape of a multivariable function by plotting it in three dimensions. This is particularly useful for understanding how the function varies not just in terms of height but also across lengths and breadths.
For our function, the 3D surface graph would illustrate a bell-like shape centered at the origin, thanks to the exponential term. Here's how it helps:

• The elevation shows the function value, offering insights into maxima and minima.
• The surface graph highlights any asymmetries or distortions caused by the polynomial part \(2x - y^2\).

Using tools like Matplotlib, students can grasp the dynamic nature of multivariable functions with more clarity by rotating and zooming into the surface to examine different perspectives.

Multivariable Functions

Multivariable functions are functions with more than one input variable, such as \(f(x, y)\). These functions map a pair of inputs to a single output, and they are crucial in fields like physics and economics where variables interact intricately.
In our function:

• The term \(2x - y^2\) is a polynomial affecting the shape of the graph.
• The term \(\exp(-x^2 - y^2)\) introduces exponential decay, centralizing the function around the origin.

Understanding these components is essential, as different parts of the function can have varied effects on graph shape and behavior. Seeing how changes in one variable affect the output can deepen your insight into real-world phenomena modeled by such functions.

Graphing Techniques

Graphing is an important skill in understanding multivariable calculus. It involves several techniques to accurately visualize how variables behave and inter-relate. In this exercise, preparing a meshgrid was crucial. This grid provides a framework of \(x, y\) coordinates over the stipulated range to compute function values effectively.
Key techniques include:

• Using mesh grids to systematically evaluate the function across a domain.
• Using libraries like Matplotlib for visual representation.

These techniques ensure not only the correct graphing of functions but also enhance the discovery of patterns and behaviors in multivariable calculus problems, making the complex terrain of these functions approachable and understandable.