Does anybody know how the graph look like?
Sketch the graph of y=e^x as a surface in R^3.
It looks the same in R3 as it does in R2. It is just the curve y = e^x, extended in both directions in the z direction. That is, it looks like a curved wall. Technically, it is an open cylinder, of exponential cross-section.
To sketch the graph of the function y = e^x as a surface in 3D, we need to consider the Cartesian coordinate system (x, y, z) where the x-coordinate represents the domain values, the y-coordinate represents the function values, and the z-coordinate represents the height or elevation of the surface.
1. Start by choosing values for x. Let's choose a range, say -3 to 3, and pick several x-values within this range. For example, x = -3, -2, -1, 0, 1, 2, 3.
2. Calculate the corresponding y-values for each x-value using the function y = e^x. For instance, for x = -3, y = e^(-3) ≈ 0.0498. Similarly, for x = -2, y = e^(-2) ≈ 0.1353 and so on.
3. Now, we have a set of (x, y) points. Plot these points on the x-y plane.
4. To visualize the third dimension, z, which represents the height or elevation of the surface, we can set the z-coordinate equal to the corresponding y-value. This means that for each (x, y) point, the z-coordinate will be the same as the y-coordinate. For example, for x = -3, y = 0.0498, the point (-3, 0.0498, 0.0498) will represent the height of the surface at that location.
5. Connect the plotted points to form a smooth curve. This curve represents the graph of y = e^x in 3D space.
6. Extend the curve along the y-axis to create a surface by replicating the same curve at different heights (z-values) along the z-axis.
7. The resulting surface will resemble a three-dimensional "bowl" opening upward, with the bottom of the bowl touching the x-y plane.
Please note that the sketch might not be perfect due to limitations in textual representation. It's recommended to refer to a graphing software or tool to get a more accurate visualization.
To sketch the graph of y = e^x as a surface in R^3, we need to represent it as a function of two variables, x and y, rather than just x.
Let's define z = e^x as a new variable. Now we have two equations: y = z and z = e^x.
To visualize this graph, we can create a three-dimensional coordinate system with x, y, and z axes. Start by choosing a range of values for x and z, such as -3 to 3 for both variables.
Now, for each combination of x and z within this range, calculate the corresponding value of y using the equation y = z.
Once you have calculated the values of y, x, and z, plot them as points in the three-dimensional coordinate system. Connect these points to form a surface.
Since y = z, the graph will take the form of a surface that rises exponentially along the x-axis. As x increases (or decreases), z also increases exponentially, resulting in an upward-curving surface.
To get a better visual representation, you can use graphing software or online tools that allow you to plot three-dimensional surfaces. These tools will provide a more accurate and detailed graph of y = e^x as a surface in R^3.
Note: If you only want to sketch the two-dimensional graph of y = e^x on the xy-plane, you can simply plot the points (x, e^x) for various values of x. This will result in a curve that increases exponentially as x increases.