Problem 71 - Calculus Early Transcendentals

Question

Use a computer to investigate the family of functions \(f(x, y)=e^{c x^{2}+y^{2}} .\) How does the shape of the graph depend on \(c ?\)

Step-by-Step Solution

Answer

The graph's shape depends on \(c\): it decays rapidly for large positive \(c\), forms a symmetric Gaussian for \(c = 0\), and elongates along the \(x\)-axis for negative \(c\).

Step 1: Understanding the Function

The given function is a multi-variable function: \( f(x, y) = e^{cx^2 + y^2} \). This is an exponential function where the exponent includes two variables, \(x\) and \(y\), with a parameter \(c\) that influences the term \(cx^2\).

Step 2: Explore the Effects of Parameter \(c\)

The parameter \(c\) modifies the stretch or compression of the function along the \(x\)-axis. If \(c > 0\), the function decreases more rapidly as \(x\) becomes large due to the positivity in \(e^{cx^2}\). If \(c < 0\), the \(x\)-contribution diminishes the exponential decay for large \(x\) values.

Step 3: Plotting for Various Values of \(c\)

Utilize graphing software or Python to plot the surface of \(f(x, y)\) for different values of \(c\), such as \(c = -1, 0, 1\). Observe the shape: - For \(c = 0\), the function resembles a Gaussian along \(y\) only. - For \(c = 1\), the function is an isotropic Gaussian (symmetric along both axes) decaying faster in both dimensions. - For \(c < 0\), the graph peaks along the \(x\)-direction rather than diminishing.

Step 4: Analyzing the Results

The graph's shape is significantly influenced by the value of \(c\): - A positive \(c\) results in a more rapid decay in all directions, creating a steep bell shape. - A zero \(c\) implies symmetry mainly affects the \(y\)-axis, and - A negative \(c\) leads to uneven shapes where the graph is less steep in the \(x\)-direction and extends out further.

Key Concepts

Exponential Functions

An exponential function is a type of mathematical function that grows or decays at a constant rate. It's characterized by the fact that the variable appears in the exponent, indicating rapid growth or decay. In multivariable calculus, these functions extend to two or more variables, creating surfaces rather than simple curves. The function given in the exercise, \( f(x, y) = e^{cx^2 + y^2} \), is a classic example, combining two variables, \(x\) and \(y\), with a parameter \(c\).

• The base here, \(e\), is a constant approximated by 2.718, known as Euler's number.
• The exponent determines the behavior of the function due to \(cx^2 + y^2\).
• In multivariable context, changes in both \(x\) and \(y\) affect the value of the function, creating a complex surface.

Understanding this helps comprehend how changes in parameters and inputs affect the overall graphical representation.

Graphing in Python

Visualizing mathematical functions using Python offers a powerful way to understand complex relationships. Tools like Matplotlib or NumPy can help create 3D plots, making it easier to see how functions behave.To graph the function \( f(x, y) = e^{cx^2 + y^2} \) for various values of \(c\):

• Use NumPy to create arrays for \(x\) and \(y\) values across a specified range.
• Compute the corresponding \(z\) values by applying the function to each point.
• With Matplotlib, use its plotting capabilities to visualize these \(x, y, z\) dimensions on a 3D surface.

This process allows you to compare different shapes that emerge from changing \(c\) and observe how they influence the surface.

Parameter Effects on Functions

Parameters in multivariable functions profoundly influence their shape and behavior. In our case, the parameter \(c\) leads to distinct surface shapes for the function \( f(x, y) = e^{cx^2 + y^2} \).

• When \(c = 0\), the shape predominantly reflects the \(y\)-axis behavior, appearing as a standard Gaussian distribution.
• With \(c \gt 0\), the function's surface becomes more steeply contoured, as the \(x\)-axis' contribution increases, leading to faster decaying in all directions.
• If \(c \lt 0\), the exponent \(cx^2\) reduces the effective growth of the function along the \(x\)-direction, which results in a depression instead of a decay, changing the symmetry and stretching the graph.

By investigating these effects, you gain insights into how altering parameters can transform a function's graphical and analytical properties.