Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ y=e^{2 z} $$

Step-by-Step Solution

Answer

The graph is a series of vertical planes, each with an exponential curve \(y = e^{2z}\), extending along the \(x\)-axis.

Step 1: Understand the Equation

The equation given is in three variables: \(x\), \(y\), and \(z\). It is \(y = e^{2z}\). Here \(y\) is expressed in terms of \(z\), and \(x\) does not appear in the equation, indicating the surface is independent of \(x\).

Step 2: Analyze the Equation

For a fixed \(z\), \(y\) equals \(e^{2z}\). This suggests that in any cross-section parallel to the \(yz\)-plane (i.e., any plane where \(x\) is a constant), the curve will be an exponential growth curve, starting above the \(z\)-axis and increasing as \(z\) increases.

Step 3: Sketch the Graph

Since \(x\) is not constrained by the equation, we can assume that the surface extends along the entire \(x\)-axis. As such, the surface consists of vertical planes stacked infinitely along the \(x\)-axis, each with an exponentially increasing curve \(y = e^{2z}\) in the \(yz\)-plane. The graph is a "sheaf" of exponential growth curves along the \(x\)-axis.

Key Concepts

Exploring Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the exercise, we encounter the function \( y = e^{2z} \). Here, the base \( e \) represents Euler's number, approximately 2.718, and \( z \) is the variable exponent. This setup describes how \( y \) changes as \( z \) changes.

This type of function is characterized by rapid increases as the variable in the exponent increases, representing exponential growth. For our particular function, as \( z \) becomes larger, \( y \) grows extremely quickly, given by the factor of \( 2z \) in the exponent. Therefore, every increase in \( z \) results in a significant multiplication in \( y \), creating a steep upward curve in graphs.

• Exponential functions display continuous growth.
• They are used to model real-world scenarios such as population growth and radioactive decay.
• The growth rate is proportional to the value of the function itself, causing the curve to get steeper rapidly.

Visualizing Three-Dimensional Surfaces

The equation \( y = e^{2z} \) helps us visualize a three-dimensional surface, even if it might seem initially challenging. Imagine the surface extending infinitely along one axis—in this case, the \( x \)-axis.

This means that for each value \( x \), the exponential curve in the \( yz \)-plane stays the same. Each cross-section parallel to the \( yz \)-plane reveals a standard exponential curve.

Visualizing a single two-dimensional curve growing steeply across the third dimension forms a surface that looks like multiple sheets or "sheaves" layered along the \( x \)-axis. This visualization can help us grasp more complicated three-dimensional shapes by considering simpler two-dimensional sections.

• A three-dimensional surface can be thought of as a collection of similar two-dimensional curves.
• The surface is represented in coordinate space, meaning it extends across different coordinate planes.
• Surfaces have different appearances based on which cross-section we view.

Understanding Coordinate Planes

Coordinate planes form the foundation of three-dimensional graphing, serving as a reference for positions and structures in space. The exercise primarily involves the \( yz \)-plane, where the equation \( y = e^{2z} \) is considered while \( x \) remains constant.

Coordinate planes are imaginary flat surfaces created by two axes that intersect at a right angle. In three-dimensional space, we often refer to the \( xy \)-plane, \( yz \)-plane, and \( zx \)-plane.

Understanding these helps us to locate and describe points and shapes in space. As seen in the exercise, each coordinate plane offers a different perspective of the graph. The ability to set one variable at a constant (like \( x \) here) simplifies complex equations into a series of manageable two-dimensional slices, offering detailed insights into the behavior of mathematical functions.

• Coordinate planes are helpful for visualizing components of three-dimensional surfaces.
• They help break down complicated structures into simpler, easier-to-understand forms.
• Each plane provides a unique view of the equation's behavior over different ranges.