Chugging along through calculus topics, we find ourselves this week looking at the level curves of a surface. First we want to realize that when you take a function with two variables, you are graphing in the 3D world. For example, any function $f(x,y)$ would need to be graphed in 3D space. Now, level curves go hand in hand with this 3D space. Basically, we want to eliminate a variable, so we can graph this instead in the 2D plane. We do this by introducing a $ \ k \ $ such that $k = f(x,y)$, where $k$ is what the function equals at some $x$ and $y$. We then solve this equation for $y$ instead, thus having an equation with only one variable. Now, graphing this equation will give you some representation of the original 3D shape in the 2D world.
A Starting Example:
Let's talk about how this actually works. For simplicity sake, take the function $f(x,y) = x + y$. From our discussion, we want to say that this function equals some $k$, so $k = x + y$. Moreover, this then just says that $y = k - x$. By graphing this equation for different values of $k$, we basically sketch out how the original function would look like in the 2D world! Below you will find what the original function looks like in the 3D world, along with the level curves! In this example, I took $ -1 \leq k \leq 5$, so we can have some ideas as to what is happening!
Level Curves:
Plane in 3D Space:
Our Function For Exploring:
Have you ever been eating your tacos and wished you did not have to lay it down on your plate? Those square base taco shells can be expensive! So, what if we just made our own taco shell holder?! In honor of national Taco day, we will construct a taco stand! We will create this shape on the 3D plane and use level curves to see what it would look like in 2D! This function serves as a nice example as to what we are wanting to accomplish as well as having a tie to (what should be considered) a global holiday!
For our function, we have: $f(x,y) = 4\cos(20x) - 2\cos(80y)$! Going again by our previous discussions, we want to make this a function with a singular variable and just change the values for $k$! Now, how do we solve this for $k$? We have a good amount of algebra and trig ahead of us!
$k = 4\cos(20x) - 2\cos(80y)$
$2\cos(80y) = 4\cos(20x) - k$
$\cos(80y) = \frac{4\cos(20x) - k}{2}$
$80y = \cos^{-1}(\frac{4\cos(20x) - k}{2})$
$y = \frac{\cos^{-1}(\frac{4\cos(20x) - k}{2})}{80}$
Now we have this nice little function in terms of just the variable $x$ and the $k$ that we will assign different values to! In order to make sure we have enough to get a sense of what the object looks like and how it behaves, I will show for when $ -4 \leq k \leq 4$. Below you will find the graph of these level curves on the 2D plane!
Level Curves For Our Taco Holder:
Okay, this may be intimidating at first, but we just want to observe some important pieces of this graph. Firstly, we see that there are some open spaces between the valleys. Also, these valleys are not very steep, but rather close together. This is an excellent position to place a taco! We are not going to make it so the taco would be in-closed, because that would defeat the purpose! Below you will find what the 3D rendering of this shape is, where the lines along the edge are the level curves that we found! Also, I placed a holder on the bottom for either taco munching on the go or just for better balancing!
Our Taco Holder:
This example was picked for the specific reason of taking an ordinary object and seeing it as a graph or function instead! Since I am in STEM Education, these sort of examples are ideas I want to incorporate in my teaching and classrooms. For many students, math is a topic that they do not notice in their everyday lives just due to past teaching methods. By finding small examples like the taco holder, I hope to bridge those gaps between mathematical topics and the real world so they may see the beauty of math in their everyday lives!
A Starting Example:
Let's talk about how this actually works. For simplicity sake, take the function $f(x,y) = x + y$. From our discussion, we want to say that this function equals some $k$, so $k = x + y$. Moreover, this then just says that $y = k - x$. By graphing this equation for different values of $k$, we basically sketch out how the original function would look like in the 2D world! Below you will find what the original function looks like in the 3D world, along with the level curves! In this example, I took $ -1 \leq k \leq 5$, so we can have some ideas as to what is happening!
Level Curves:
Plane in 3D Space:
Our Function For Exploring:
Have you ever been eating your tacos and wished you did not have to lay it down on your plate? Those square base taco shells can be expensive! So, what if we just made our own taco shell holder?! In honor of national Taco day, we will construct a taco stand! We will create this shape on the 3D plane and use level curves to see what it would look like in 2D! This function serves as a nice example as to what we are wanting to accomplish as well as having a tie to (what should be considered) a global holiday!
For our function, we have: $f(x,y) = 4\cos(20x) - 2\cos(80y)$! Going again by our previous discussions, we want to make this a function with a singular variable and just change the values for $k$! Now, how do we solve this for $k$? We have a good amount of algebra and trig ahead of us!
$k = 4\cos(20x) - 2\cos(80y)$
$2\cos(80y) = 4\cos(20x) - k$
$\cos(80y) = \frac{4\cos(20x) - k}{2}$
$80y = \cos^{-1}(\frac{4\cos(20x) - k}{2})$
$y = \frac{\cos^{-1}(\frac{4\cos(20x) - k}{2})}{80}$
Now we have this nice little function in terms of just the variable $x$ and the $k$ that we will assign different values to! In order to make sure we have enough to get a sense of what the object looks like and how it behaves, I will show for when $ -4 \leq k \leq 4$. Below you will find the graph of these level curves on the 2D plane!
Level Curves For Our Taco Holder:
Okay, this may be intimidating at first, but we just want to observe some important pieces of this graph. Firstly, we see that there are some open spaces between the valleys. Also, these valleys are not very steep, but rather close together. This is an excellent position to place a taco! We are not going to make it so the taco would be in-closed, because that would defeat the purpose! Below you will find what the 3D rendering of this shape is, where the lines along the edge are the level curves that we found! Also, I placed a holder on the bottom for either taco munching on the go or just for better balancing!
Our Taco Holder:
This example was picked for the specific reason of taking an ordinary object and seeing it as a graph or function instead! Since I am in STEM Education, these sort of examples are ideas I want to incorporate in my teaching and classrooms. For many students, math is a topic that they do not notice in their everyday lives just due to past teaching methods. By finding small examples like the taco holder, I hope to bridge those gaps between mathematical topics and the real world so they may see the beauty of math in their everyday lives!
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