which-point-on-the-curve-ysqrtx-21-is-closest-to-the-point-41-what-is-this-minimum-distance-2

Name: which point on the curve ysqrtx 21 is closest to the point 41 what is this minimum distance 2
Uploaded: 2020-07-29
Duration: 1 min 46 s
Channel: Numerade
Description: Which point on the curve $y=\sqrt{x-2}+1$ is closest to the point (4,1)$?$ What is this minimum distance?

Question

Which point on the curve $y=\sqrt{x-2}+1$ is closest to the point (4,1)$?$ What is this minimum distance?

Sriram Soundarrajan · Accepted Answer

you hear the function by easy kowtow route off Express to bless one. And the point is for what we had to fight. Medium distance tragical to this point. So there Girl Point can be found on by days ago. Toe off X minus for the host card. That&#x27;s by minus one. The horse quit, so he&#x27;s a little group off into squared minus eight X plus 60 plus by square minus two way glass. Mark Deasy Curto rule. It&#x27;s a square minus seven. Next, plus 40 is by using a corridor into square minus seven, next for dinner. So if we differentiate by a nice affects, which can be correct my nursery it goes little exactly girdle sung by by using quarto, certain Babel Plus two Bressler, which is included a tow find toe. So and you&#x27;re just a point is really quite faith. Gonna find it through boot and their distance. The is a total of 3.5 square minus seven or three point fight Bless 14. It&#x27;s comes to be 1.3

Ankit Gupta · Answer

for the graph off. Why equals X Plus one? We need to find the point which is closes to the point for one. So let&#x27;s plot the graph. Find the boy now on the graph. The 0.41 is here. So the point which will be closest to the point for one lying on the graph point which will be at the perpendicular distance off X units on that point, is 23 So the closest point is to treat toe for one on the graph for vise equal to X plus one.

Matthew Lee · Answer

So you know it. We need to find a function where, Trevor, than the distance between point on a curve and the point they&#x27;re coming to get one now. Any pointing a curve? Coordinate X comma. I squared my too. And now that we have 2.3 canopied a distant one month D did you or choir room back mining? You think this is our personal second point? The first coin. Excellent job square. What? And then square my two money negative one. Nothing squared. All right, I&#x27;m making simplify that to make and then you&#x27;re going to be X squared. Uh, mining one squared. All right, You&#x27;re the function of X. Um, I gotta square root. And to make this the competition a little bit here, you better make movies are function. We&#x27;re going to make d squared are functions we escorted side of the equation. We&#x27;ll get, uh, square X squared, minus one. Where and this we&#x27;re going to make we&#x27;re going to call it f of X now. So if we minimise at vexed, then for minimizing this B squared square, didn&#x27;t But when the square of the beautiful minimize the This is awesome. In a mine. So that&#x27;s okay. We can just minimize the simpler functions. And now we need upon the minimum the function. Uh, I&#x27;d like to find the derivative like you could for a derivative test to find the minimum before you do that, though might simplify the function. So expand out the term we get next group quiet. Except for my two X squared clash one. When combining terms, we get extra before my ex crim one. All right. And now we&#x27;re ready to find a minimum of dysfunction. Take the derivative. We&#x27;ll do the first relative test we need to find where the derivative equal to zero. So the derivative is the function for him. My direction and next, like, finally, it&#x27;s derivative equal to zero so that it equals zero for executing on a direction. And then this offer actress factor. So, Nero, you go factor on eggs. But I wasn&#x27;t even talking to exterior. We will get Greg my one and now get to the ball Inclusion to actually pull a girl or the following equation. Brother Practical. They&#x27;re out and two X squared minus one equals zero. And here forget I equals zero. It&#x27;s a critical point And here I saw for X and take the square root. So we&#x27;ll get after my Gregory, Uh, 1/2 which could be rational. I, uh, like your mind, uh, squared of you over too. All right, here are 33 critical numbers in critical point and the fish to figure out which one of the minimum. That&#x27;s quite a first derivative. Uh huh. We&#x27;re going to set up a number line for the value of driven it when we have euro. You&#x27;re an incredible number of your hero. Question. My reach over to that would put her. Told me to be here somewhere. Well, negativity will be too very somewhere, and we need to find out when the derivative positive and one negative. Uh, so you critical number. They&#x27;re going to break break up our number one in the interval. And so to figure out the door of a positive or negative somewhere next point into the derivative. So I took the point on here. It could be any point. So much of you 10. Kind of. Definitely bigger. Require October tail. You&#x27;re a fucking 10 into the derivative. Here. We&#x27;ll get four time. Really? Our number, uh, 5000 minus 20. Definitely a positive number. Derivative is positive here. And then somebody will be here in, like, a fucking negative 10. So negative, uh, we&#x27;ll get a really large negative number. Negative point outing that my next to time. Take your time. Pointed a negative number. All right. Next. You can try number in here so we could do, like, negative 0.1, for example. And if you the calculator negative 0.1, you&#x27;ll find a derivative positive. You&#x27;re and over here could plug in positive 0.1, and you&#x27;ll find a derivative negative. All right, So the minimum of a function occurs when the derivative goes from positive to negative and look like this. Excuse me. Uh, negative. Positive affected. So visualize the minimum value. Look, I&#x27;ve drawn it wrong. Thanks, man. I&#x27;m saying the correct word, but I&#x27;m sure the wrong picture. So from negative to positive, I mean, their function decreasing angry thing like this off. It&#x27;s negative, goes up, goes down and the negative positive come off. So so? And so we see that negative were to over two. One positive. It&#x27;s over to These are both minimum value throwing ex Nichols what? Their money makeover too. The function minimize. Remember, this function represents it doesn&#x27;t square. That means for these acts value. This is between the curve and the point of being minimized. Now, we&#x27;re not done. We need to find the actual point on a curve. Well, we just started echoed by you. So I just plug back into this equation for the curb to find out why that you. So we dug in, um, re plugging routes over to function. Do you, like, with a record amount of two, and we&#x27;ll get, uh, to over four. Oh, they did more than think I&#x27;m actually, uh, when we No matter, we need a positive or the negative. We&#x27;re going to get a positive. The fourth minus two, which turned out to be negative three house. That&#x27;s going to be this. Whether or not we&#x27;re talking a positive printer, beauty or negative, it over to that number is being squared. All right, so let&#x27;s give it a two point. They have negative. You over too. Promise. Negative rehab. Now the positive rich over too. Do we have need of a two point on the curve, like with great to my two, which are closest to the 0.0

Chris Trentman · Answer

whereas to find the point on the curve Y equals the square root of X, it is closest to the 0.30 being able to be humiliated. Well, what is the distance from this curve to this point? This distance, which are called D as a function of X and Y. It&#x27;s the square root of X minus three squared plus y minus zero squared, which is the square root of X minus three squared plus y squared. This is the same as just X, I think part every assaulted. Now we know that our distance D is going to be minimized when function F, which is D squared, is minimized as well. This is because D is a increasing positive function, so F as a function of X. This is D as a function of X squared, which is X minus three squared plus X now to find the minimum of effort going to take the derivative F prime of X, just two times X minus three plus one, find critical values to set this equal to zero. Yeah, so we have X minus three equals negative. One half, or X equals negative one half plus three, which is, I&#x27;m still positive. Five hats. Yeah, this one. Actually, this was now the second derivative F double prime of X. This is equal to to which is, of course, always positive. Therefore, it follows that our function f has an absolute minimum at its critical value. Well, it has an absolute minimum on its domain, from zero to infinity. Well, actually notes it&#x27;s on all and he got his son who is living and breathing on all real numbers X. And it has it at the critical value. X equals five halves. Yeah, inside yourself. So the point that&#x27;s closest has coordinates X equals five halves. And why is the square root of five halves There&#x27;s no Freddy hurt? Yes, it&#x27;s which, of course, is the same as the square root of 10 over to. And so we have the points five halves. Route 10 over to this is the point closest to the 0.0.30 on the curve. White will swear to vex, so mm

Linh Vu · Answer

So it just for my weekend right down. Is that want you go, Joe, Do you want us to x So a born Isn&#x27;t that bond on the graph? Uh, I&#x27;m just grab has a formal Uh huh. On June honest to x amount of distance. We&#x27;re conditioned with me that big a distant from be ju A whole regions and a former now de equal to the square root of the X one is so square Thio honest, Your X men is their square When because of the tradition is you Can you Could you like square plus too honest you Thanks. Square on when you&#x27;re minimalist system will be this I mean you My, uh, x square plus doesn&#x27;t can expand. It was you end up for minus eight extra place for X square. You need your money minus one on. Yeah, we will. Judge your Rio Grande is one in June. Nice phone will be Isn&#x27;t will be five x square when it&#x27;s eight angelesfour and work that it goes end upon months. Being with you may so good. I am driven by Jordan&#x27;s 5 to 10 for under five on this one would be, uh, along with being a minimized. This is a little bit. Yeah, and we&#x27;re going to accept. Yeah, go on him. We could You&#x27;re far five. And that one will be jewels minus d&#x27;oh terms for under five. So that&#x27;s going really good. You found five b. Do you remind us I want five and you&#x27;re going to Ah d&#x27;oh! Under five rejoinder here. And this is exactly what we&#x27;re looking for because it&#x27;s true, Arjun.

Taylor Shimono · Answer

So for this problem, we can go ahead and begin by setting up a diagram of our unit. Circle X squared plus y squared is equal to one. So we&#x27;ll go ahead and just make a graph and say that it&#x27;s going to look approximately like this, but the radius of one and its center of the origin. And we want to know the closest point on this graph to the point to one which is going to be right here. And so we know that we&#x27;re going to be able to use the distance formula and we could see that it you look about our line. Thio point. You look about like this. I&#x27;m back from the distance formula. We know that the distance squared is equal to our difference in exes. So X minus two scored close our difference. In wise, we could be y minus one&#x27;s word. And so I&#x27;m actually going to set this out this distance squared equal to length l because it&#x27;s not going to make a difference in our, uh, evaluation of where our minimum or maximum will be. It&#x27;s only going to you change. Um, the fact that the length is equal to the distance squared, but it&#x27;s just going to make it a little bit easier for us to calculate our derivative. So if we distribute out each of these terms on the right, we get our length is going to be equal to X squared minus for X minus four plus y squared minus two wife plus one. And if we sub student from our equation, our original equation that were given and we know that why squared is equal to one minus X squared. So that means that we can write this as our length is equal to X squared minus four X minus four plus y squared, which is equal to one minus expert minus two. Why we know that Y is equal to the square root of one minus X word plus one. And so when we simplify and take the derivative, we&#x27;re going to end up finding my deal. The ex I should be equal to a negative for minus two times negative to X over two times the square root of one minus X squared, and when we plug, Seth is equal to zero and sold for ex Rianna. Finding that X is equal to either the positive or negative to over this crew of five. And so when we look at her diagram, um, it&#x27;s pretty clear that the X value is going to have to be positive here. So that means that we&#x27;re just going to take positive again and solve for the Y value. This will correspond to by plugging into our original equation. I&#x27;m so to over the square to five squared it&#x27;s going to be forfeits. Plus y squared is equal to one, so you find that this is going to correspond to wise equal to one over the square to five. So our point under graph that&#x27;s going to be close is 2 to 1 is the 0.2 over the square to five one over the square to five.

Problem

Find the coordinates of the point on the line $y=…

Problem 13 Easy Difficulty

Which point on the curve $y=\sqrt{x-2}+1$ is closest to the point (4,1)$?$ What is this minimum distance?

Answer

$(3.5,2.22), 1.32$

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David Cohen, Theodore B. Lee, David Sklar

Precalculus With Unit-Circle Trigonometry

Grace H.

Anna Marie V.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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$(3.5,2.22), 1.32$

Precalculus With Unit-Circle Trigonometry

Grace H.

Anna Marie V.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

David Cohen, Theodore B. Lee, David SklarPrecalculus With Unit-Circle Trigonometry

Grace H.

Anna Marie V.

Kristen K.

Michael J.

David Cohen, Theodore B. Lee, David Sklar

Precalculus With Unit-Circle Trigonometry