Problem 34 - Calculus Early Transcendentals

Question

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=e^{-x^{2}-2 y^{2}} ;[-2,2] \times[-2,2].$$

Step-by-Step Solution

Answer

Question: Graph two level curves for the function \(z=e^{-x^{2}-2y^{2}}\) within the window \([-2, 2] \times [-2, 2]\). Label the level curves with their corresponding \(z\)-values.

Step 1: Identify the given function and window

The given function is \(z=e^{-x^{2}-2y^{2}}\). We want to graph the level curves within the window \([-2, 2] \times [-2, 2]\), which means \(x\) varies from \(-2\) to \(2\) and \(y\) varies from \(-2\) to \(2\).

Step 2: Set z as a constant

To find the level curves, we need to set \(z\) as a constant value, say \(c\). So we have \(c=e^{-x^{2}-2y^{2}}\).

Step 3: Solve for y

We want to solve for \(y\), so we can find the function of the level curves. We can rewrite the equation as: $$y=\pm\sqrt{-\frac{1}{2}\ln(c)-\frac{x^2}{2}}$$

Step 4: Choose two z-values

Now, let's choose two \(z\)-values, say \(c_1 = 0.5\) and \(c_2 = 0.2\). These \(z\)-values will give us two level curves that we will graph.

Step 5: Find level curves for chosen z-values

Substitute the chosen \(z\)-values to the equation from step 3. For \(c_1 = 0.5\): $$y=\pm\sqrt{-\frac{1}{2}\ln(0.5)-\frac{x^2}{2}}$$ For \(c_2 = 0.2\): $$y=\pm\sqrt{-\frac{1}{2}\ln(0.2)-\frac{x^2}{2}}$$

Step 6: Graph the level curves

Now we graph the level curves within the window \([-2, 2] \times [-2, 2]\). Plot the two level curve functions we found in step 5 in the window defined. Make sure the chosen level curves are labeled with their respective \(z\)-values (\(c_1 = 0.5\) and \(c_2 = 0.2\)). At this point, the graph should show two level curves for the function \(z=e^{-x^{2}-2y^{2}}\) within the specified window, labeled with their corresponding \(z\)-values.

Key Concepts

Contour Plot

A contour plot is a graphical representation used to show level curves of a function with two variables. In a contour plot, each line represents points where the function has the same value, known as the "contour level." These plots are especially useful in multivariable calculus as they offer a way to visualize complex three-dimensional surfaces on a two-dimensional plane.
Contour plots are powerful tools:

• They help understand regions of the function where values are constant.
• They simplify the analysis of functions by examining specific levels.
• They can be applied easily to scenarios like topographical maps.

To create a contour plot, identify the function and set it to different constant values, then solve for the variables at those levels. This way, you can draw the contours to get insights into the behavior of the function.

Exponential Function

The exponential function, denoted by functions that have a constant base raised to a variable exponent, is one of the most important mathematical functions. In the exercise, the function given is an exponential function of the form \[ z=e^{-x^2-2y^2} \] featuring a negative exponent. It is characterized by how rapidly it increases or decreases and is frequently used to model growth and decay processes in nature and science.

• The exponential function is continuous and smooth, making it easily graphable.
• When the exponent involves squares of variables like here, it often forms bell-shaped curves known as Gaussian functions.
• It is essential in probability, statistics, and fields involving dynamic changes over time.

Understanding how to work with exponential functions is critical as they appear everywhere, from describing phenomena in physics to financial modeling.

Multivariable Calculus

In multivariable calculus, we handle functions of two or more variables. Unlike single-variable calculus, which focuses on changing quantities over one dimension, multivariable calculus explores how functions behave with many inputs.
Here are some key aspects of multivariable calculus related to level curves:

• It involves finding partial derivatives to understand the function's rate of change in multiple directions.
• Level curves are a foundational concept where the function's value is constant.
• It requires new graphical tools like contour plots to analyze and interpret data over two dimensions.

Through multivariable calculus, one can study more complex systems and solve real-world problems involving several changing quantities simultaneously, such as temperature variation, pressure fields, and economic models.

Graphing Techniques

Graphing techniques in mathematics involve drawing functions or data points on a coordinate plane to analyze and interpret them visually. The key steps to graphing involve understanding the function's behavior, setting boundaries, and marking important values accurately.
For the function \[ z=e^{-x^2-2y^2} \]in the problem, essential steps include:

• Setting a plotting window, in this case, \([-2, 2] \times [-2, 2]\).
• Choosing specific contour values like \(z=0.5\) and \(z=0.2\) to draw level curves.
• Using graphing software or plotting by hand, ensure each contour line is accurately labeled with the correct z-value.

Graphing effectively communicates the function's behavior and relationships between variables, making it an indispensable tool in calculus and other mathematical fields.