{Use of Tech} Special curves The following classical curves ha...

Question

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x⁴ - x² + y² = 0 (Figure-8 curve)

x⁴ - x² + y² = 0 (Figure-8 curve)

Accepted Answer

Hello. In this video, we are going to be graphing the given classical curve using analytical methods, including implicit differentiation. So the function given to us is X to the 4th power minus 4 X2 + 4Y2 equal to 0. So, in order to graph this function, there's a few things we first need to find. First, let's go ahead and talk about the functions domain. Now, what we have here is a function in terms of X and Y, but what's important to note is that the function is in the form of a polynomial. Polynomials have a domain of all real numbers, therefore, the domain of this function is going to be all real numbers from negative infinity to positive infinity. Now what we want to do is you want to calculate the Y intercept of the graph. In order to calculate the Y intercept, we want to set the variable X equal to 0 in the given equation. When we do this, we are left with 0 to 4th power which is 0, minus 4 multiplied by 02, which is 0, plus 4 Y2 equal to 0. This will simplify to be 4 Y2 equal to 0, and if we divide by 4 and take the square root, we are left with Y is equal to 0. What this means is that the location of the Y intercept of this function is going to be at the origin, 0.0. So we will go ahead and label that point on our graph. Now what we want to do is you want to go ahead and calculate the X intercepts. Now, in order to calculate the X intercept, we will be setting the variable Y equal to 0. Now, that will leave us with X to the 4th power minus 4 X2 plus 42, which is 0, equal to 0. Now what we need to do is we need to solve for X. We can start by factoring out an X2 from each term. That will leave us with X2 multiplied by X2 minus 4, equal to 0. If we set both factors of X equal to 0, we get two equations, X2 equal to 0 and X2 minus 4 equal to 0. The first equation will simplified to be X is equal to 0, indicating that we have an X intercept at the origin, and that is the same location as the Y intercept. Now, on the right-hand equation, we will add 4 to both sides and take the square root, and that will leave us with X is equal to plus or minus 2. So we have two X intercepts other than the origin, one located at 2.0 and the other located at -2.0. Let's go ahead and label these points on our graph. Now that we've calculated the intercepts and the domain, let's go ahead and discuss the symmetry of this function. Now, let's go ahead and check to see if the function has symmetry about the Y axis. In order to determine symmetry about the y axis, we want to determine if the function is even or odd. So, in order to determine whether the function has symmetry but the y axis, we will go ahead and plug in the value of negative X into our function. That is going to be F of negative X equal. Actually, not F of X. This is a function in terms of X and Y. So actually, when we plug in negative X, we will be left with negative X to the 4th power. Minus 4 multiplied by negative X2, plus 4 Y2 equal to 0. Note that any time we raise a negative number to an even power, that will simplify to be a positive value. So this is going to simplify to be X the 4th power minus 4 X2 + 4 Y2 equal to 0, which is the original function. What this means is that F of negative X is equal to the original function. Therefore, we have symmetry about the y axis. Now we need to check to see if we have symmetry about the X axis. In order to check symmetry about the X-axis, we will plug in the value. Of negative Y into our function. Doing so, we'll we can rewrite our function as X minus 4 X2 plus 4 multiplied by negative Y2 equal to 0. By the same reasoning as the previous step, negative Y2 will simplify to be Y2, which will further simplify to be the original function. So that means that when we plug in the value of negative Y, F of negative Y is equal to the original function. Therefore, this implies that we have symmetry about the X axis. Finally, we need to check to see if we have symmetry about the line Y is equal to X. In order to check this, we will plug in the values of negative Y and X into our function. But notice that from the from the previous two steps, if we do this, we will end up with the original function, thus implying that we have symmetry about the line Y is equal to X. So, our graph is completely symmetric. Now that we've discussed the symmetry, let's go ahead and calculate the critical points of the function. In order to calculate the critical points, we will need to take the derivative of the function. Since we have a function in terms of X and Y, we will go ahead and take the derivative with respect to X, and use our methods of implicit differentiation. So, that is going to give us D over DX, so the derivative of respect to X of X to the 4th power minus 4 X2 + 4 Y2 equal to 0. By using the power rule, the derivative of X to the 4th will be 4 X cubed. The derivative of -4X2 will be -8 X, and the derivative of 4Y2 will be 8Y. But because of implicit differentiation, since we are taking a derivative of Y with respect to X, we will go ahead and generate a DYDX term, and this is equal to the derivative of zero, which is 0. Now what we want to do is we want to solve for the derivative DYDX. We will go ahead and add 8X to the right-hand side, and -4 X cube to the left-hand side, allowing us to rewrite the function as 8YDYDX. Equal to 8 X minus 4 X cubed. Notice that each coefficient has a multiple of 4, or is a multiple of 4. We can divide the entire equation by 4, leaving us with 2 Y D YDX equal to 2 X minus X cubed. And if we divide both sides of this equation by 2 Y, we have D Y D X equal to 2 X minus X cubed divided by 2 Y. This is going to be the derivative of the original function with respect to X. Now what we want to do is we want to find the locations of the critical points. In order to do this, we want to find the values of X that caused the numerator to be zero. So we will take our numerator, 2 X minus X cubed, set it equal to 0, and solve for X. We can factor out an X from this equation, leaving us with X multiplied by 2 minus X squared. We will set both factors of X equal to 0, leaving us with Xs equal to 0 and 2 minus X2 equal to 0. So, we have one critical point at X is equal to 0, and on the right hand side, we will go ahead and add X2 to both sides of the equation, leaving us with X2 equal to 2. And when we take the square root, we have X's equal to positive and negative square root of 2, which is going to be two more critical points. So, we know that X equal to 0 is also the X intercept. So we know that this is located at the origin because the X and Y intercept are at the same points. But what we need to do now is we need to determine the locations of the other critical points, positive square root of 2, and negative square root of 2. In order to determine these locations, we can take these values and plug it back into the original function. So, if we were to take square root of 2 and plug it into the original function, we will get the value of positive and negative 1. And if we were to take negative square root of 2, and plug it into the function, we will also get the value of positive and negative 1. What this means is that both of the critical points produce two values of the graph, which we will go ahead and label. So, we can approximate square root of 2 to be roughly around here. So we have positive and -1 as our Y values. The same can be applied for negative square root or 2 at positive and -1. And finally, if we graph this function with respect to our symmetries, the final version of our graph is roughly going to look like an infinity symbol. And this is going to be the rough sketch of the original function. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.x⁴ - x² + y² = 0 (Figure-8 curve)

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{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x⁴ - x² + y² = 0 (Figure-8 curve)