Problem 43 - Calculus Early Transcendentals

Question

Draw a contour map of the function showing several level curves. $$f(x, y)=y e^{x}$$

Step-by-Step Solution

Answer

Plot level curves for \( y = \frac{c}{e^x} \) to create a contour map, choosing multiple values for \( c \).

Step 1: Understand the Function

The function given is \( f(x, y) = y e^{x} \). This is a function of two variables, \( x \) and \( y \). The exponential function \( e^x \) grows as \( x \) increases, and \( y \) scales this growth.

Step 2: Define Level Curves

Level curves are curves along which the function \( f(x, y) \) is constant. This means for a given constant \( c \), we have the equation \( f(x, y) = c \). Substituting the given function, we get \( y e^x = c \), which simplifies to \( y = \frac{c}{e^x} \).

Step 3: Solve for y in Terms of x and c

From \( y = \frac{c}{e^x} \), we see that for various values of \( c \), \( y \) changes with respect to \( x \). These equations represent different level curves of the function.

Step 4: Plot the Level Curves

Choose different constant values for \( c \), such as \( c = 1, 2, 3, -1, -2, -3 \) to plot several level curves. For each \( c \), graph \( y = \frac{c}{e^x} \) on a coordinate plane with \( x \,\) versus \( y \).

Step 5: Draw the Contour Map

Using the plotted level curves, draw the contour map. The contour map will consist of these curves on the \( x-y \) plane, where each curve indicates a different constant value of the function \( f(x, y) = y e^x \).

Key Concepts

Function of Two Variables

In calculus, a function of two variables is an essential concept. It means the function depends on two inputs or variables. For our exercise, the function is given as \( f(x, y) = y e^{x} \). Here, both \( x \) and \( y \) are independent variables, and \( e^{x} \) is the exponential part that affects the behavior of the function. The output or result of the function depends on the values we choose for these variables.
Understanding this concept is crucial before we can move on to drawing the contour map. For any combination of \( x \) and \( y \), we calculate \( f(x, y) \) to determine the function’s value. This reflects how the function behaves across the plane where both \( x \) and \( y \) vary.

• Think of it like a surface hovering above the \( x-y \) plane.
• The height of this surface at any point \((x, y)\) is given by \( f(x, y) \).

This concept sets the stage for understanding more complex topics such as contour maps and level curves.

Level Curves

Level curves are an integral part of understanding functions of two variables. They represent places on the \( x-y \) plane where the function has the same value. For the function \( f(x, y) = y e^x \), a level curve for a constant \( c \) is defined by the equation \( y e^x = c \). Solving for \( y \), we have \( y = \frac{c}{e^x} \). These equations give the different level curves of the function.
Level curves translate the 3D surface perspective to a 2D plane, making it easier to visualize. Think of them like contour lines on a topographical map that indicate areas of equal elevation. Here's how to approach them:

• Choose specific values for \( c \), such as 1, 2, 3, -1, etc.
• For each \( c \), graph the equation \( y = \frac{c}{e^x} \) on the \( x-y \) plane.

This will help visualize the contour map, with each line representing a different constant value of the given function.

Exponential Function

The exponential function \( e^x \) is a fundamental mathematical function. It's characterized by rapid growth as \( x \) increases, making it impactful in various fields like finance, biology, and physics. Here, in the function \( f(x, y) = y e^x \), the exponential part \( e^x \) contributes significantly to the overall behavior of the function.
When considering the function’s impact on level curves, the exponential element controls how \( y \) changes in response to \( x \). With a higher \( x \), \( e^x \) grows, and \( y \) needs to decrease for the product \( y e^x \) to remain constant for a given \( c \). Conversely, for negative \( x \), \( e^x \) decreases, allowing \( y \) to increase. This dynamic showcases the exponential function's unique role in shaping the appearance of level curves. Understanding the exponential growth and decay helps in predicting not just the behavior of the level curves but also in anticipating how the function behaves across plane changes. This knowledge forms a critical base for further exploration in calculus, particularly in evaluating more complex relationships between variables.