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Chapter 9: Problem 8

In Exercises \(1-8,\) describe and sketch the surface. $$ z-e^{y}=0 $$

Short Answer

Expert verified
The surface \(z = e^{y}\) is a vertical 'bedsheet' that extends infinitely along the x-axis, it rises steeply for positive \(y\) values and smoothly levels off towards the xy-plane as \(y\) goes negative.

Step by step solution

01

Analyze the given function

The given function is \(z = e^{y}\). This means that the value of \(z\) depends only on \(y\), irrespective of the value of \(x\). In 3D space, an increase or decrease in \(y\) affects \(z\) but changing \(x\) has no effect on \(z\). Hence, the surface will be a pattern which is repeated along the x-axis.
02

Understand the behavior of the function

We know that the exponential function \(e^{y}\) is always positive and it increases as \(y\) increases. When \(y = 0\), \(z = 1\); as \(y\) approaches infinity, \(z\) also approaches infinity, and as \(y\) approaches negative infinity, \(z\) approaches 0 but never reaches 0. This tells us that the surface will rise steeply for positive \(y\) and level off to the xy-plane as \(y\) becomes negative.
03

Sketch the graph

Using the information we have gathered from the previous steps, we can sketch the surface. Start by plotting the curve \(z = e^{y}\) in the yz-plane (letting \(x = 0\)), which gives a rising exponential curve. Next, replicate this curve along the x-axis. This will create a series of vertical lines parallel to the x-axis. The overall surface appears as an infinitely long 'bedsheet' that rises steeply from the xy-plane for positive \(y\) values and smoothly levels off towards the xy-plane as \(y\) goes negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function
The exponential function, typically written as \(e^{y}\) or \(e^{x}\), is a mathematical expression that features prominently in many areas of calculus and applied mathematics. The function's base, \(e\), is an irrational number approximately equal to 2.71828. One key characteristic of the exponential function is that its value grows rapidly as the input value (in this case, \(y\)) increases.
This growth can be summarized as follows:
  • For positive \(y\), \(e^{y}\) is large and increases swiftly.
  • For \(y = 0\), \(e^{y} = 1\).
  • For negative \(y\), \(e^{y}\) approaches 0 but never becomes negative or zero.
This function models phenomena in natural sciences where growth processes are exponential, like population growth, radioactive decay, and interest compounding.
3D Graphing
Graphing in three dimensions can seem tricky, but it's all about understanding how functions behave with respect to three axes: \(x\), \(y\), and \(z\). When we graph the equation \(z = e^{y}\), we're looking at how the values of \(z\) (vertical height) change corresponding to changes in \(y\), while maintaining that \(x\) doesn't influence \(z\).
Here's a simplified approach to understanding 3D graphing for this exercise:
  • Begin by considering the 2D plot of \(z = e^{y}\) in the yz-plane. This results in a curve that rises exponentially as \(y\) increases and approaches the axis as \(y\) decreases.
  • Since \(x\) doesn't affect \(z\), extend this curve along the \(x\)-axis. Imagine carrying forward the same curve infinitely in both positive and negative \(x\) directions.
  • The resulting shape is an infinite surface, resembling a 'sheet', stretching along the \(x\)-axis. This visual helps to understand the relationship between the geometry of the curve and the volume it occupies in 3D space.
Coordinate Planes
The discussion of coordinate planes is essential when visualizing surfaces in three-dimensional space. The coordinate system consists of three perpendicular lines or planes, which help in identifying locations in 3D space. Each point is represented by a triplet \((x, y, z)\).
There are three primary planes:
  • **xy-plane**: A flat surface defined by the \(x\) and \(y\) coordinates, with \(z = 0\). This plane spreads horizontally and is often seen as the floor or base of a 3D graph.
  • **yz-plane**: This incorporates \(y\) and \(z\) axes, where \(x = 0\). For our exercise, this plane is where the basic exponential curve (\(z = e^{y}\)) is initially drawn.
  • **xz-plane**: Defined by \(x\) and \(z\), assuming \(y = 0\). This plane is vertical and allows visualization of changes not dependent on \(y\).
Understanding these planes gives a framework to map and visualize 3-dimensional functions clearly. They help in isolating behavior of functions like \(z = e^{y}\), making it easier to comprehend how changes in each variable impacts the graph.

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