find-the-curvature-kappa-for-the-curve-yx-frac14-x2-at-the-point-x2

Name: find the curvature kappa for the curve yx frac14 x2 at the point x2
Uploaded: 2020-04-30
Duration: 1 min 57 s
Channel: Numerade
Description: Find the curvature $\kappa$ for the curve $y=x-\frac{1}{4} x^{2}$ at the point $x=2$ .

Question

Find the curvature $\kappa$ for the curve $y=x-\frac{1}{4} x^{2}$ at the point $x=2$ .

Linh Vu · Accepted Answer

function y equals ju X minus or not far thanks square and the question asked her to find it curvature onda value up to so we know done the curvature k we know in the game of the exit with Iko June the absolute no wide about graham the Veniamin one plus y prime square Totally about under a tree and a Jew And therefore we need to find the Boston Mubi, the white Bram where it goes You were given the one here. The father What chemical? To one minus 1/2 on the X. And then the one of a family could humanise half. And therefore we have everything If when we have the guy next week Oh, joke. What about removing 1/2? Dividing by, uh is absolutely so we have again the 1/2 now Now we will have one breast the one minus 1/2 square Everything about our ambulatory under two. So we on, uh, the nominee they nominated. We can certainly find this one and then we get go We don&#x27;t need to signify because from here we just need to look in a very okay Joe him fo kept you. We go tell 1/2 inviting by. We haven&#x27;t had to be one. Plus, you go with her to inside. Here we get the one minus one B zero B, one breath. So, you know, about 300 to 4. We get equal to one on with Julie over one and go to another to we know final answer here.

Amy Jiang · Answer

Okay, so we&#x27;re giving the function. Why? Executive one over three execute. No, I&#x27;m asked to find in the curvature. So we take, um, first orbit of it in the second. So we get X squared for the first second derivative. Is this two x curvature? See if X is equal to why does time at X and one plus white crime wrecks Too hard too. Andrea. Over too. To put mess that and we get this is equal to two X over extra for this one. For you too. Not like me in X is equal to one. We get to over 23 or two, which simplifies discovered or two over.

Jack Chen · Answer

for this problem. We have a functionary off X equals two X cubed plus two X minus one. So first we need to compute the first of the narrative and the second on the directive, you know, little to figure out the curvature now when X&#x27;d close to two. We have the first of the purity of you posting 14 and the state conduct narrative because to 12th means about a formula off the curvature. You coast to the absolute value of F double front divided by one process. Trying square to the three half that gives us 12 over 1 97 to 3 helps no.

Wen Zheng · Answer

it was a graphing calculator or a computer. It&#x27;s your grass. Both the Kouf and is a watcher function on the same screen. So why is he going to axe to four minus two times X squared? First we can compute a white problem. It&#x27;s equal for X Cube, minus four acts. Why Crumb from physical? Twelve pounds, exclaimed minus four. Then we&#x27;re half the overture. Is culture absolutely off? Why? From from over One bus y prom squire, the power ofthe really want to. This is the absolute volume. Twelve exclaim Linus, or over one glass? Well, ax cube minus tax. She&#x27;s a power off. Very hungry, too. Now let&#x27;s look at it to graph this curve on this coercion is that right? Curve is a curve. Off is a graph with proof and a blue is a graph of the coercion

Tanishq Gupta · Answer

Yeah, but I didn&#x27;t. Yes. Get off. Why Awful. Yeah, From the excess. Thanks, Dash Existing? Yes. No, that five x four. So maybe more defy different by next. So this is, I think.

Anna Marie Vagnozzi · Answer

all right. In this problem, we want to be able to graph the function y equals X to the fourth power minus two times X squared and the curvature of that function on the same graph window. So that way we can compare the two and see if we can get an idea from the graph of why How to predict what the curvature graphite look like. So first thing we need to do is actually find function for the curvature. What I&#x27;m gonna do is I&#x27;m gonna use the formula in in the box in the top right corner. So I&#x27;m going to need the first and second derivatives of why, with respect to X. So what will get for the first derivative is for execute minus for X and for the second derivative get 12 times X squared, minus four. So then we know that the curvature Kappa of X going to be equal to the absolute value of why double prime. So 12 X squared, minus four over one plus Why? Prime squared, sir. Four x cubed minus for X squared. All raised Teoh the three house power. And we don&#x27;t need to really simplify this very much since we&#x27;re gonna plug it into the calculator. So first, what we&#x27;ll do is we will look at the graph of why so what I&#x27;ll do is and I have a t I 84 calculator. All go to my graph menu. Why one clear anything that&#x27;s in in there. So in 11 I&#x27;m going to put my function for why? So we&#x27;re gonna have X rays to the fourth power minus two x squared, and we&#x27;re gonna go ahead and graph that I&#x27;ve already messed around with the window a little bit so you can see what works best for you. But I found it easiest to view when we have arrange from negative 5 to 5 on our X coordinates and from negative 2 to 8 on our white coordinates. Then I&#x27;m gonna click grass so you can see what that looks like. All right. And remember, when we&#x27;re looking at this, that the curvature at a point is how quickly the graph is changing directions at that point. So if I look at the curve that we get in our in our graphing calculator, we&#x27;ve seen it. There are three inflection points where this graph is changing direction or turning around. So we might expect for the curvature to be greatest where those points happen. So what we could say is that we expect to see the greatest curvature. No, at the inflection points of why, In other words, if we&#x27;re looking for the greatest curvature, that means that we would expect Kappa of X or curvature function to have ah, relative Maxima at those inflection points. And we can see if we look at a graph we put our, uh, ex And why ticks facing at just one So we can see that are inflection points occur at X equals negative 10 and one. Now, let&#x27;s see if we&#x27;re right. What I&#x27;m gonna do is I&#x27;m gonna go back to my y equals menu. I&#x27;m gonna go toe. Why, too? So that way we can leave our original function on the graph, and I&#x27;m going to use my left Cherokee to hover over that, uh, that how bar and I&#x27;m gonna click. Enter once and you can see that it&#x27;s changed from a thin bar to a thick bar. So that&#x27;s just gonna change our line style. So that way we can make sure we see the graph clearly in distinction from the graph of why So what we&#x27;re gonna do is we&#x27;re gonna put in our function for capita Vex. That&#x27;s gonna be the absolute value, which I can find him there. My math numbers meant you and we&#x27;re gonna have 12 x squared minus for that&#x27;s in the numerator. We&#x27;re gonna divide that by one plus the quantity four x cubed minus for X but squared clears off those parentheses and that denominators raised to the power of three over to. And then I&#x27;m gonna click graphic, and what we&#x27;re gonna see is the graph of our curvature also show up. And if we look closely, we can see that does have three relative Maxima. And those are indeed at those points where we have inflection on why so we have relative Maxima at X equals negative one X equals zero and what so this is how we can think about the relationship between the graph and a curve, and the graph of that curves curvature by considering where those points of I haven&#x27;t. But the greatest change in direction occur which correspond to the greatest curvature and relative maximum on our on a graph of capital fix

Problem

Find the curvature $\kappa$ for the curve $y=\fra…

Problem 133 Hard Difficulty

Find the curvature $\kappa$ for the curve $y=x-\frac{1}{4} x^{2}$ at the point $x=2$ .

Answer

$\kappa(2)=\frac{1}{2}$

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