Explore Level Curves of $f(x,y)=x^3-x$

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In summary, the level curve is the intersection of the horizontal line $y=c$ with the curve $y=x^3-x-c$. If the polynomial has three roots, the level curve consists of three lines. If the polynomial has a double root, the level curve consists of two lines. And if the polynomial has non-real roots, the level curve consists of one line.
  • #1
mathmari
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Hey! :eek:

I have to describe the behaviour, while c is changing, of the level curve $f(x,y)=c$ for the function $f(x,y)=x^3-x$.

I have done the following:

The level curves are defined by $$\{(x,y)\mid x^3-x=c\}$$

For $c=0$ we have that the set consists of the lines $x=0,x=1,x=-1$.

Is it correct so far?? (Wondering)

How could we continue ?? What can we say about the other values if $c$?? Which is the set when $c$ is positiv and which when $c$ is negative?? (Wondering)
 
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  • #2
Hi,

Plot $x^3-x$.

Geometrically, what you are doing is drawing a vertical line $x=whatever$ at the intersection points of an horizontal line $y=c$ with the curve $y=x^3-x$.

You will got as many vertical lines as intersection points you have with the horizontal one, taking a look at the plot you can easily figure out what happens, and then try to describe it analitically (if needed).
 
  • #3
I got stuck right now...

Do I not have to take cases for $c$ ($c=0$, $c<0$, $c>0$) and find which the set of level curves is described?? (Wondering)

Or am I supposed to do something else?? (Wondering)
 
  • #4
The level curves are defined by the set $L_c=\{(x,y)\mid x^3-x=c\}$.

The function $f(x,y)$ depends only on $x$. So, if $(x_0,y_0)\in L_c$, then $(x_0, y)\in L_c, \forall y\in \mathbb{R}$.

Therefore, each level set $\{f(x,y)=c\}$ is the union of the lines $\{x=x_0\}$ in the plane, where $x_0$ belongs to the set of roots of $f(x,y)-c=x^3-x-c$.

Since this a cubic polynomial, depending on the value of $c$, it can have three real simple roots, one real simple root and one real double root or one real simple root and two non-real roots, right? (Wondering)
Do we describe in that way the behaviour of the level curve? (Wondering) When the polynomial has three roots does the level curve consist of three lines? (Wondering)

What happens when the polynomial has a double root? (Wondering)

And what happens when it has non-real roots? (Wondering) Also what information do we get for the graph of $f$ ? (Wondering)
 

Related to Explore Level Curves of $f(x,y)=x^3-x$

What is the function f(x,y)=x^3-x?

The function f(x,y)=x^3-x is a mathematical expression that represents a relationship between two variables, x and y. It is a polynomial function with a degree of 3, meaning that the highest exponent of x is 3. This function can be graphed in a two-dimensional coordinate system, with x and y as the axes, and it produces a three-dimensional surface.

What are level curves of f(x,y)=x^3-x?

Level curves, also known as contour lines, are a type of curve that represents points on a surface with the same function value. In the case of f(x,y)=x^3-x, the level curves would be curves that connect points with the same function value, or z-value, on the three-dimensional surface. These curves can be plotted in a two-dimensional coordinate system, with x and y as the axes, and they can help visualize the shape of the surface.

How do I graph the level curves of f(x,y)=x^3-x?

To graph the level curves of f(x,y)=x^3-x, you can use a computer program or graphing calculator that allows you to input the function and plot the curves. You can also manually plot the curves by choosing different values for x and y, calculating the corresponding z-value using the function, and then plotting the points on a graph. By connecting these points, you can create the level curves.

What do level curves tell us about the function f(x,y)=x^3-x?

Level curves can provide information about the behavior of the function f(x,y)=x^3-x. For example, if the level curves are close together, it indicates that the function is changing rapidly in that area. On the other hand, if the level curves are far apart, it suggests that the function is changing slowly. Additionally, the shape of the level curves can reveal the overall shape of the surface and any patterns or symmetries within the function.

What is the significance of exploring level curves of f(x,y)=x^3-x?

Exploring the level curves of f(x,y)=x^3-x can help us better understand the behavior of this function and its relationship with the variables x and y. It can also aid in visualizing the shape of the surface and identifying any critical points, such as local maxima or minima. This information can be useful in various fields of science, including physics, economics, and engineering.

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