Question

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

Short answer

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

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.

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.

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.