Introduction
Level curves can help us visualize a 3-dimensional object on the 2-dimensional plane. The most common example that you might have seen before is a topographic map. In the image below without realizing you may be able to tell how this graph might look in 3-dimensions. We can see that there are multiple lines, and we can notice that they do not cross each other. Each line that you see represents a different elevation or height which is why the lines do not intersect. Another thing you might have noticed is that the closer together the lines are the steeper the slope would be. This is how we can represent a mountain on a piece of paper.
Example
Let's look at an example to see how this works. We will be looking at the function \[f(x,y)=\frac{0.1x^2+0.1y^2}{2}\] Then I have further bounded this by \begin{align*} -10≤x≤10 \\ -10≤y≤10 \end{align*}
Our function that I have chosen is paraboloid but then is cut off making a square bowl. This would give us a shape of a square bowl like the image on the right. We can see how the height and the shape of the bowl changes. The slope of the bowl is also getting steeper as we come near the edge of the bowl.
As you may have guessed we are going to use level curves to represent this bowl on the xy-plane. To find a level curve we will be using k-values that equals the function we are using: \[k=\frac{0.1x^2+0.1y^2}{2}\]
Here if we needed to we could have solved y or x if it could have made it easier to visualize how the curves would look like. However, if we keep it equal to k we can already see how the curves would look like because of the simple function we chose. For each k-value we give to our function would be a different level curve. For example, if k=1 we would find all the x and y values that make this function equal 1 and since the chosen function is a paraboloid, we know the level curve would look like an ellipse. Thanks to GeoGebra we can easily find the level curves of the function which can be seen below.
In the image above we see the level curves of our function. We are able to tell as we increase the k-value the level curves are getting closer to each other. By this we are able to tell the slope is increasing. These are the level curves to the function, but we also have x and y bounds which would change how our level curves would look. Below we can see how the bounds affect the level curves. The distance from one level curve to the next is still the same but the length of the curve we see is what changes.
The level curves in the image above are what they would look like for the square bowl we have. With the image below we can see how the bowl looks like with the level curves on it. We can see where the level curves that we calculated would appear on the bowl. There are 10 level curves that can be seen on the bowl which is minimized to 2X2 inches.
Why this example
By using this example, it makes it easier for new learners to understand this topic. By being able to relate the image to something that they have seen or can visualize can help them go through the thought process. I chose to use a simpler function because by doing this I did not have to solve for y or x because we already knew how the function would look like allowing more focus on the topic. We are still able to see how the curves would be closer together when the function gets steeper. Also by adding bounds I was able to show that the level curves can be broken even though they have the same height.
Level curves can help us visualize a 3-dimensional object on the 2-dimensional plane. The most common example that you might have seen before is a topographic map. In the image below without realizing you may be able to tell how this graph might look in 3-dimensions. We can see that there are multiple lines, and we can notice that they do not cross each other. Each line that you see represents a different elevation or height which is why the lines do not intersect. Another thing you might have noticed is that the closer together the lines are the steeper the slope would be. This is how we can represent a mountain on a piece of paper.
Example
Let's look at an example to see how this works. We will be looking at the function \[f(x,y)=\frac{0.1x^2+0.1y^2}{2}\] Then I have further bounded this by \begin{align*} -10≤x≤10 \\ -10≤y≤10 \end{align*}
Our function that I have chosen is paraboloid but then is cut off making a square bowl. This would give us a shape of a square bowl like the image on the right. We can see how the height and the shape of the bowl changes. The slope of the bowl is also getting steeper as we come near the edge of the bowl.
As you may have guessed we are going to use level curves to represent this bowl on the xy-plane. To find a level curve we will be using k-values that equals the function we are using: \[k=\frac{0.1x^2+0.1y^2}{2}\]
Here if we needed to we could have solved y or x if it could have made it easier to visualize how the curves would look like. However, if we keep it equal to k we can already see how the curves would look like because of the simple function we chose. For each k-value we give to our function would be a different level curve. For example, if k=1 we would find all the x and y values that make this function equal 1 and since the chosen function is a paraboloid, we know the level curve would look like an ellipse. Thanks to GeoGebra we can easily find the level curves of the function which can be seen below.
In the image above we see the level curves of our function. We are able to tell as we increase the k-value the level curves are getting closer to each other. By this we are able to tell the slope is increasing. These are the level curves to the function, but we also have x and y bounds which would change how our level curves would look. Below we can see how the bounds affect the level curves. The distance from one level curve to the next is still the same but the length of the curve we see is what changes.
The level curves in the image above are what they would look like for the square bowl we have. With the image below we can see how the bowl looks like with the level curves on it. We can see where the level curves that we calculated would appear on the bowl. There are 10 level curves that can be seen on the bowl which is minimized to 2X2 inches.
Why this example
By using this example, it makes it easier for new learners to understand this topic. By being able to relate the image to something that they have seen or can visualize can help them go through the thought process. I chose to use a simpler function because by doing this I did not have to solve for y or x because we already knew how the function would look like allowing more focus on the topic. We are still able to see how the curves would be closer together when the function gets steeper. Also by adding bounds I was able to show that the level curves can be broken even though they have the same height.
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