{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.

y = 8/(x² + 4) (Witch of Agnesi)

y = 8/(x² + 4) (Witch of Agnesi)

Accepted Answer

Welcome back, everyone. In this problem, we want to graph the given classical curve, the Witch of Agnesi, Y equals 27 divided by X2 + 9, using analytical methods, which includes implicit differentiation. Here we have a graph on which we graph the which of Agnesy. Now, if we're going to graph or curve, there are a few properties that we'll need, OK? So let's try to figure out those properties through the analytical methods. And the first thing we may want to think about is symmetry, OK? Now, we know that this graph as a classical curve is symmetric about the Y axis, OK? So let's make a note of that. It's already symmetric above the y-axis. Now, we'll need to start thinking about intercepts. If we look at our graph. Or if we look at our equation Y equals X2 + 9, in this case, if we think about it here, if we want to find a value. OK, that's going to make Y equal to 0. There will be no value because if X is squared is going to be positive, thus our answer is always going to be greater than 9. It's going to be 9 plus something. So we know our graph has no intercepts. Let's also make a note of that. No, in terms of why intercepts, to find a Y intercept, we just need to substitute. Y to be equal to 0, OK. So for the Y intercept our substitute X to be equal to 0. The Y value is going to be equal to 27 divided by 02 + 9, which equals 27 divided by 9 or 3. Thus, the Y intercept is going to be 03, OK? Let's let's make a note of that. Next, we need to figure out uh the asymptotes for our graph. No, for the for its asymptodes. OK. And we know it, it will have asymptotes based on the form of the equation. Then we know that from looking at our equation, as X approaches infinity, OK? In other words, as X gets bigger, Y will approach 0, OK. And as X approaches. Well, here, that's pretty much it actually. As it approaches infinity then Y will approach 0 on both sides really because remember this is gonna be a squared term. So we know that there is a horizontal asymptote. A horizontal asymptote. At the point where Y is equal to 0, it will continue to approach that, OK? So now that we have the asymptos, the intercepts, the symmetry, next, let's start to talk about the um critical points. And to do that, we'll need to differentiate our graph. Now, we can implicitly differentiate it. So now we're trying to find our critical points. And to find our critical points, we can differentiate our equation and set it to zero. So no. Given why it was. 27 divided by x2 + 9 we could rewrite it as y multiplied by. Y multiplied by x2 + 9 equal to 27 and know if we differentiate both sides with respect to x on the left hand side we can apply the product rule to get the derivative of Y multiplied by X2 + 9 plus Y multiplied by the derivative of X2 + 9, which is 2 X. And that will be equal to 0. So, no, this tells us that the derivative of Y will be equal to -2 XY, OK, divided by X2 + 9. And since we know Y equals 27 divided by X2 + 9, if we substitute that in the expression, then we get the derivative to be equal to negative 54 X divided by X2 + 9 all squared. No, like we said at this point. The critical point will be where a derivative is equal to 0, OK? So at the critical point. OK. Our derivative equals 0. And if we set this entire expression to 0, then we know that our numerator -54 X will be equal to 0, and thus, that means X equals 0. Therefore, 03. Is the critical point of our graph. No, next we can find our inflection points. And we know that our inflection points are at the point where the 2nd derivative equals 0. What's the second derivative of our function? Well, we already have the 1st derivative, OK, so our second derivative. It is going to be equal to. The derivative with respect to X of 54 X divided by X2 + 92. And now we know to figure out this derivative, we can use the quotient rule and the possibility the chain rule of differentiation. And when we do that. Then we should find that the second derivative equals 162 multiplied by X2 minus 3. Divided by X2 + 9 raised to the 3. Like we said, at the inflection point, our second derivative is 0, OK? So at the inflection points. The second derivative equals 0. So now if we set this to 0, then we should get that X2 minus 3 equals 0, which means that X will be equal to + R minus root 3. No, what are our corresponding Y values here? Well, when X equals root 3, OK, then Y equals 27 divided by root 32 + 9, which equals 27 divided by 12 or 2.25. Therefore, The inflection points. are at Route 3. 2.25. And negative route 3 2.25. And we are sure that this would, this negative, this value would work for this negative value of X, negative root 3. Because remember that if we put negative root 3 here and square root, we would still get the same answer as root 3 squared. So now let's just do a quick check of all the information that we have, OK? Now, so far, we know that our inflection points are Route 3 2.25 and negative route 3, 2.25. We know that 03 is the critical point. We know that there's a horizontal asymptote at Y equals 0, and we know the Y intercept is 03, while there are no X intercepts and that our graph is symmetric about the axis. So, let's plot our known points. So we have our Y intercept, 03, uh, or inflection points. So that's gonna be uh 2.25 negative root 3. It's gonna be probably somewhere right here, right? And then we have 2. Sorry, Route 3 2.25. So that's gonna be somewhere here. Let me fix that a little bit. Just a little bit here and then a little bit out here, OK. I think I'm not. Let me just uh fix this a little bit here. My apologies. So we have Route 3 here and then we're also gonna have Route 3 here, OK. And now we know we have our horizontal asymptotes, so we can do our curve. So we know our curve is gonna look something like this, OK? Because remember, our horizontal asymptote is at Y equals 0. And this is going to be the graph of the Witch of Agnesy. Thanks a lot for watching everyone. I hope this video helped.

{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.y = 8/(x² + 4) (Witch of Agnesi)

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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.

y = 8/(x² + 4) (Witch of Agnesi)