{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.

x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)

x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says graph the given classical curve using analytical methods, including implicit differentiation. We're given the equation X to the 2/3 power plus Y 2/3 power is equal to 16. Below the problem we're given an empty graph on which to plot our function. Now, the curve that we're given here in this problem is a super ellipse. So it's going to look like a rectangle with rounded edges. That also means that our function here is going to be symmetric about both the X and Y axis. So the first thing we need to do is actually find the intercepts along the X and Y axis. So, we're gonna start off by looking at the Y intercept. So, we'll set X equal to 0. In doing so, we're going to have the 2/3 power is equal to 16. And so we can solve this for Y. Doing so, we'll have Y is equal to plus or minus. 16 raised to the 3 halves power. Which is going to be equal to plus or minus 64. So that'll give us two intercepts on the Y axis, and if we look at the X-axis intercepts, we'll set Y equal to 0, then we'll have X raised to the 2/3 power. is equal to 16. Solving for X will have X is equal to plus or minus 16 raised to the three halves power. Which is equal to plus or minus 64. So, we can go ahead and plot those points on our graph. Along the X and Y axis. And here you do see that we do have that. Symmetry Across both the X and Y axis. So now what we need to do is calculate the first derivative, so the derivative of Y with respect to X, and for this we're going to use implicit differentiation. So we'll take the derivative with respect to X. Of the quantity of both sides of our equation here. And so we'll have X raised to the 2/3 power, plus Y raised to the 2/3 power in quantity, and that's going to be equal to the derivative with respect to X of 16. So taking the derivative on the left hand side, we're gonna need to use our power rule. Take this derivative. So for the first term, we'll have 2 divided by 3, multiplied by x raised to the minus 1/3 power. Then we'll have less for the second term. We will have 2 divided by 3, multiplied by Y raised to the minus 1/3 power, multiplied by the derivative of Y with respect to X since we since we have to apply our chain rule as well here. For that second term, and this is going to be equal to 0, since the derivative of a constant is equal to 0. So we want to solve this for the derivative of Y with respect to X. The first thing I notice is that I can divide both of the both sides of this equation by 2 divided by 3, and so that will get rid of the factor out in front of both terms on the left hand side. Then I subtract both sides by X raised to the minus 1/3 power. And so I'll have 1 divided by Y raised to the 1/3 power, multiplied by the derivative of Y with respect to X. And that's going to be equal to -1 divided by X raised to the 1/3 power. So here we just rewritten X and Y raised to the minus 1/3 power as 1 divided by those variables raised to the 1/3 power. So now I just need to multiply both sides of the equation by Y raises to the 1/3 power. And I'll get the derivative of Y with respect to X equal to minus Y raised to the 1/3 power divided by X raised to the 1/3. So when both X and Y are positive, that means our derivative here is going to be negative, so our function should be decreasing in the first quadrant. So now we have enough information to draw this graph. Remembering that we are dealing with a super ellipse, so that means that we are drawing a box here, in this case a square. That has rounded edges so we're going to connect. Are intercepts. With rounded edges, and so in our first quadrant here, our function is going to be decreasing. And then we can just use symmetry to draw the remaining quadrants. And so, in the 2nd quadrant, our function is also going to be decreasing. When going for or it's actually going to be increasing starting from R X intercept and going to our Y intercept. In the third quadrant, our function will be decreasing going from the X intercept to the Y intercept. And then our function will be increasing. In the 4th quadrant, going from the Y intercept to the X intercept value. So, what we have drawn on the screen here is the answer to this question. So, I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Implicit Differentiation

Graphing Utility

Astroid

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