use-a-graphing-utility-to-generate-the-curve-yx3-left1x2right-in-the-x-y-plane-and-then-use-the-grap

Name: use a graphing utility to generate the curve yx3 left1x2right in the x y plane and then use the grap
Uploaded: 2020-09-03
Duration: 39 s
Channel: Numerade
Description: Use a graphing utility to generate the curve $y=x^{3} /\left(1+x^{2}\right)$ in the $x y$ -plane, and then use the graph to help sketch the surface $z=y^{3} /\left(1+y^{2}\right)$ in 3 -space.

Question

Use a graphing utility to generate the curve $y=x^{3} /\left(1+x^{2}\right)$ in the $x y$ -plane, and then use the graph to help sketch the surface $z=y^{3} /\left(1+y^{2}\right)$ in 3 -space.

Linh Vu · Accepted Answer

in this question, use on Greta and ah question were given. It&#x27;s the ICO junior wine Power three and then be riding by No. One Plus why square and as we look and the figure the picture here will say it is about allowing to the excesses it&#x27;s we expect, because the variable X is not in the question here.

Linh Vu · Answer

in this question would you use on a couple cackle? Rita? And a question with even is Z equals June. Why? Being finding by one blast? Why power far. And here as we expect the graph the surface here we&#x27;re but allowed tuna excesses because the variable X is not in the question. So does we. Made it grabbed? Looks like here.

Linh Vu · Answer

use on a cat. Rita, in the equation given is one equal chew one minus. Why out you? So we say we should have the proble here, but it would be the food plan that will be cuffed down at this is gonna be a in a treaty. Craft looks like here.

Patrick Vaughn · Answer

and this problem, we want to find the well. We want to sketch the surface z equal to the square root of one plus x squared plus y squared. So for this problem, we&#x27;re gonna need to recall the different kinds of quadratic services. Um, here are the six kind of basic forms that will be playing with So we&#x27;re gonna want to identify which one of these this belongs Thio. And we&#x27;re gonna need to remember how we sketch these using traces and stuff like that. Kind of what they look like. Okay, So if I was to square both sides of this equation, I would get Z squared is equal to one plus X squared plus y squared so I&#x27;d have Z squared minus X squared minus y squared is equal to one. So this is a hyper Floyd of two sheets where a, B and C are equal to one. So we&#x27;re gonna have Z is greater than or equal to zero. But so if I was to plug in zero for X and wide, get Z is equal to one, so we can just say it&#x27;s strictly greater than zero, or we&#x27;re gonna say it&#x27;s greater than equal to one. That&#x27;s fine too. Okay, so now we wanna recall what the shape of, ah to sheet hyperbole It is. So it has for Texas, uh, above and below the origin. And then we get these little bull shapes inside of them with ellipses for cross sections for different levels of the okay. And since we&#x27;re just looking at positive values, is he? We can just throw out all this on the bottom if we sketch this craft. So we need our axes. Okay, move that There, I suppose. And we&#x27;ll need some traces. So when Z is equal to zero, we have negative X squared minus y squared is equal to once. This isn&#x27;t what we want we want to do is pick like Z equal to two right now. Let&#x27;s dio Yeah, he was fine, so this would be minus two squared. So we get X squared minus y squared equals plus y squared is equal to three. So that&#x27;s Ah, circle with radius equals the square to three. Right, So here&#x27;s one and two, and then we&#x27;re gonna want So what is it when z is equal to zero now? When When X equals zero We have Z squared minus X squared is equal to one. So that&#x27;s the hyper Bola. And then if y is equal to zero, we got Z squared. Minus y squared is equal Thio One. Okay, so we have thes high privileges in the X Y plane and xz plane. Gosh, you cannot seem to get that curve right There we go experience that&#x27;s in my Z y plane. And then let&#x27;s do the easy explains Well, and then we have these circular cross sections or different values of Z. So, like, for instance, here&#x27;s to right, so we&#x27;ll plot these points. So here&#x27;s a point. There is a point. There&#x27;s a point and there&#x27;s a point. So that&#x27;s so here. The vertex right is going to be 001 and then on this circle, we&#x27;re gonna have so here. Zero um, the radius was square to three. Right to and negative. Route three 02 zero Negative. Route 32 And up here we have fruit. 302 Alright, that completes our sketch

Patrick Vaughn · Answer

and this problem were given the quadratic service. Z equals the square root of X squared plus y squared minus one. I wanna sketch it. So first, let&#x27;s side to try to identify what kind of service this is. So if I was to square both sides of my equation, I would get Z squared equals X squared plus y squared minus one. Now, if I move the one toe So let&#x27;s move these to the left hand side and then multiply everything by negative one So I&#x27;ll have negative Z squared plus X squared plus y squared is equal to one. Now, if we remember from this section the different kinds of quadratic surfaces will notice that this is ah hyperba Lloyd of one sheet because we have a negative coefficient frenzy and positive coefficients in front of X and y so a B and C would be equal toe want. Um, we have one on the right hand side. This is over, like one squared over one squared, etcetera. Oh, right. So we have ah high privilege of one sheet. However, right, if we were to solve for Z, we would get Z is equal to plus or minus the square of X squared plus y squared minus one. So we&#x27;re cutting out the negative component in our equation here. So we&#x27;re gonna be saying that Z is greater than or equal to zero. So we just wanna weigh. Just want to sketch Are one sheet hyper void, but, you know, honor above the X Y plant. Okay. And so what did a one sheet hyper boy looked like? It kind of has hyperba la&#x27;s, um, for traces on the the X Z and the y Z axes. And it has ellipsis on the x y um uh, X y plane So but we&#x27;re only looking at the positive and zero values of the Okay, so let&#x27;s go ahead. And we want to draw that. So we need a We need our axes already. Get this right There we go on Go. We&#x27;ll need some traces. So, for instance, if Z was zero, we have x squared. Plus y squared equals one. That&#x27;s a circle with radius one and then x zero. So it, uh, why squared minus disease squared equals wants that Sirola and X squared minus d squared equals one when white. So that&#x27;s why you will zero x equal zeros. Cool. Zero. So these are traces, right? So now we can plot some of these. So, uh, for instance, here is our circle of radius one. It&#x27;s not very good Australian. So point point, point point. Yeah, okay. And so we&#x27;re getting these high purple is coming off of this and then circles around those e No, what I want. So like, if we want to know what the radius would be, we&#x27;d have s o X squared. Plus y squared is equal to one plus z scored. So our would equal the square of one plus z squared. Alright, that completes that problem.

Carson Merrill · Answer

Okay, So what we see here is we first want to start by graphing. Why equals e to the X and we want to do it just in the are too. So that&#x27;s the plane. Just that just involves X and y. So this is the graph that we see Onda were pretty familiar with the graph of y equals e to the X Because it&#x27;s this typical exponential equation. Then we go into three dimensions and we want to know Well, what does it look like if we have y equals e to the X in this dimension in three dimensions? What we see it follows a very similar curve. If we were to just make this flat, it looks like the exact same curve that we saw before. Um, it would look something like this. However, the difference is if we move about in the third dimension, we see that it has a depth that goes as faras egos. So no matter what Z is, we have this y equals e to the X shape of the graph. But then it also has a third dimension that z next. What we want to dio is graph y equal actually easy equals. Eat the X So me grab Z equals easy X. We&#x27;re going to see something very similar. Um, the only difference is now, instead of the width of the graph being related Thio Z Now the width of the graph is related to why is we see, no matter how far, why goes we have this e to the X looking graph, but in this case it&#x27;s a function of Z, which is why it&#x27;s going up here going down here. This is changing in with regards to Z as X increases or decreases, and then the width of the graph is a result of why.

Problem

Use a graphing utility to generate the curve $y=x…

Problem 45 Medium Difficulty

Use a graphing utility to generate the curve $y=x^{3} /\left(1+x^{2}\right)$ in the $x y$ -plane, and then use the graph to help sketch the surface $z=y^{3} /\left(1+y^{2}\right)$ in 3 -space.

Answer

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