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Question

Find the point on the parabola $x=t, y=t^{2},-\infty<t<\infty$
closest to the point $(2,1 / 2) .$ (Hint: Minimize the square of the
distance as a function of $t .$ )

Luuk Verhoeven · Accepted Answer

in this question, we are given a brochure ization off the standards So x this is given by T and the corresponding white Gordon. It is C squared. So let&#x27;s make it speak, drawing what it looks like we were. And why axes? And then it just looks like this guy. That&#x27;s terrible, drawing more like that and it goes through zero. Then we&#x27;re also giving the points, too. 1/2 which is roughly here, let&#x27;s say and we&#x27;re justifying the shortest distance to this point. So, um, well, this is an optimization question so that that&#x27;s try to figure out which question we need to open eyes. Let&#x27;s first figure out the distance as a function off time. Not any distance in this case is just getting by. Um, decorous. So suppose we are over here when we get this nice writing triangle. So the distance is the distance in the extraction, which is no to minus except e weird, um, plus the distance in the wind direction, which is, uh, 1/2 minus. Why FT here? Um noted, technically, the distance in the ex direction is the absolute value off to minus X. Um, so just two minus X, But since we&#x27;re squaring it anyway, keeping track of those upbeat values, um, isn&#x27;t gonna do anything for us. So now let&#x27;s plug in. Uh, tea are in the lift. Logan are known expressions for X and y. So we have this equals two minus T squared, plus one minus. He&#x27;s weird square. Now we can work out these squares. Well, see, this will become for my for tea plus t squared. And this because a order minus steets were less You 2 to 4. So you can just check that this works out using foil, or like the normal rules from a publication. Um, now, let&#x27;s addle like terms together and organized. It&#x27;s a bit nicer, so that&#x27;s starting to deter the fourth. There&#x27;s no t cute. Um, we have plus t squared minus d squared. So there&#x27;s no teeth where it&#x27;s left over. We do have this minus 40 and then finally, we have the four plus in order. So we get, uh, this for wasn&#x27;t order and I can simplify this to 17 over four. Yeah, but we&#x27;ll leave it like this for now. So now we want to figure out, um When, uh, it is minimal. So we want to minimize the distance function. The distance is a function of time. So to do that, we computed derivative prime off T. Um, so we get for she cubes minus for, uh, zero. So 40 cute, minus fourth derivative. And since you&#x27;re optimizing the tea for which the distance takes some extreme value, so a local minimum or maximum is as a time for which interrupted zero So great d prime off t the road. Isn&#x27;t it easier if the plane we&#x27;re after is zero four times t zero pubes minus four equals zero. Well, we can divide by four and we find a tease era pubes equals one removed four to the other side before. And that tells us that in fact T zero, there&#x27;s one. So be it. Distance is minimized at time, equal to p zero. So what is this? What is the corresponding points next of T zero equals one and why off zero equals one? So this point here we were after wait 11

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.

Prathan Jarupoonphol · Answer

it&#x27;s video. We are asked to find the minimum this tens from this proble to the 0.0 tree. So we have to translate this into a function to use Ah, black grunge multiplier. The This 10 however, is you see, to understand so is like that this 10 from the origin any any point it why will be there. This 10 is ah drew it off According in square right here is almost the same. But instead off the origin we we use zero tree as a base. So our this time between parabola and this point would be this function right and is the function we want to minimize the Palavela itself is the restriction function. So we have g asked this one. And as usual we will look at the grade Ian&#x27;s equation between these two. So I have calculated this great and out like this. Dave the recording it going to tail a step. Linda is minus one And this interview tell us from D Sorry from the second Coordinate that Why, ese? So is minus one over two plus three, which is five over too. Yes. And ah, since seeing the restoration function years, why is X square. We already know that this is X square. We don&#x27;t even have to five eggs because in the in the function the function is this 10 and use X square already. Okay, so if he&#x27;s x square plus why my next three square We know just now the X square is five over To what about? Why ministry? We also know it here from here. Why ministry alone would be miners 1/2 So square it. We will get one over four. Right? This give 11 over too. So this is the extra mom off function. If on interns it will be the extra mom off the distant. But ah, sorry. Or for 11 over four. Not too. I&#x27;m sorry, but don&#x27;t forget that the this 10 is self need square room here. We don&#x27;t use square road in function f to simplify our calculation. In the end, if it acts for this tent, we have to to square root off. If so, that this time would be Squire rode off. You live in over four. Which is if you want to break it. So choir Route 11 Oh two it is must be to hear not for? Because four is also under square root. And that is the answer. The minimum distended between this too. Thank you.

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

maximum vertical distance between the line white was X plus two and the parameter of y equals X squared for X between negative one and positive too. Well, to do this first, let&#x27;s find what the vertical distances between these two graphs recall. This distance D legal. There&#x27;s a function of X. Well, this is going to be the square root of the difference of why coordinates. This is X squared minus X plus two squared. Now we know the vertical distance MM is maximized when he squared is maximized as well. So do you swear, Becks? I call this function f a bex. Well, this is X squared minus expose to squared. Now, to find the maximum vertical distance forward. To find the maximum of this function f of X, we&#x27;re going to find its derivative. So at the prime of X is by the chain ruled two times X squared minus X minus two times two X minus one and we set this equal to zero. Well, this is only equal to zero when X squared minus X minus two equals zero or two x minus one equals zero. In the first case, the discriminate Delta E squared which is one minus four times a and see well, it&#x27;s positive. So it has solutions. In fact, we can factor this as X minus two times X plus one equals zero. And so we get X equals two, where X equals negative one. Her second equation. This gives us X equals one half. However, we&#x27;re only looking for excess between negative one and two. But all of these satisfy that. So we have three critical points native one one half hand, too. Now, we&#x27;re going to compare the value of F I usually these critical points so f of negative one to plug this in. This is negative one squared, which is one minus one minus two squared negative two squared or four. I&#x27;m sorry. This should be one plus one minus two squared, which is zero f of one half. This is one half squared or four minus one half minus two. This is negative. 1/4 minus two. All this squared now. This is negative. 11 4th squared, no negative. 7/4 squared, no negative 9/4 square. My mistake. So this is 81 16th. Likewise. F two plug this in. This is two square which is four minus two minus two squared. This is zero squared or zero. So it&#x27;s clear that yeah, the maximum value occurs at one of the critical values were at the endpoints, which happened to be the same as two of the critical values. And so f has absolute maximum on the interval. Negative 12 at X equals one half. Therefore, the maximum vertical distance the of one half. Well, this is the square root of one half squared, Just 1/4 minus one half minus two squared, which is the same as the absolute value. Uh huh. Negative 9/4 which is just positive. 9/4. So this is our answer.

Matthew Lee · Answer

uh, too fine. A point on the curve. Brute Can y which cooked to the point you&#x27;re coming for, uh, start by actually grabbing that perv. And we gave him intuition for it by actually stopping for Why stop there. Why excoriate? Why, Then why don&#x27;t we call it divided by 10 one or two? Next square. This is actually the equation of a problem. Um, and if we were to grab the problem would look something like this, uh, cried originally that you called a square root of 10. Why? So it means X is greater than zero. So this problem that I agree with zero that is the domain of its function, the points are calling for it. I&#x27;m sure somewhere we&#x27;re looking for the point on the problem on this problem, which couldn&#x27;t this point. They&#x27;re coming for it now. An opportunity point of the problem. If we call the we&#x27;re talking, actually forget whenever 10 x squared. Is that appointing a problem that we need to minimize the distance between point that point, uh, want to write an equation for that? Using the distance. Do you want you quote, uh, that we can compute? This is our X to the X one, though we&#x27;ll have extra money. Euro squared course the difference in the wide which is one over 10 X squared, minus four. The one over 10 X squared. What for square. Okay. And whenever we want to minimize the distance, we can always just minimize. Just minimize the square of the distance because dysfunctional, be minimum a function and minimize When? If it doesn&#x27;t minimum number here and its square that number of still be a minimum. Oh, um, we can minimize d squared function We&#x27;re going to be minimizing. Will be square of this baby with you, Um, without the quarry to make it a lot easier to thought. Yeah, me too different gate and find a critical numbers. So, um, you get for reading. Simplify this a little bit. Of course. Just be extra one. Overturn. I scored my four square so we could distribute this, But go ahead and just take it. Find incredible numbers right away. Finding too derivative. Uh, the French. Is that this part of just Karol to Rex This cart, We&#x27;re gonna see the General A 12 Tirol The general, The portrait generals have different. Is this even a car roll, too. Time when you get inside the same whenever. 10 extraordinary for anymore multiplied by the derivative of the inside, which is using the car roll. Get to over 10 check and in minus in Europe. All right, get some of this a little bit of direction on course then that actually just this year Oh, we&#x27;re gonna forget about that part. Nobody in front of the earth and more to point on these terms together, we&#x27;re gonna get, uh so well, when you want a time. So driving to to get to over 10 squared money. Eight. I&#x27;m Joe over tonight, and now I&#x27;m watching you page that you called. But your evidence we drove the truck there, and then what you got? Uh, tribute this in here. We&#x27;re going to get four out of 100 x Q? Yeah. Um, minus. Make 16 over 10. Yeah, You never know this. Okay, fine. A little bit. Uh, t Rex. It is a fraction won over 25 cube, and then this is the fraction you were 10. Yeah, Conjoined. She&#x27;d like to get it over five. And then if we combine the terms to two is the same. A 10 pitch, the temp. It once it that makes you. And then we have but one over 25 cubes There. That in my favor. All right, if you question it. Yeah, Okay. Incredible. Numbered. When when you find Max, that makes the derivative hero we had recorded here on top. And do that, Uh, are you factoring out of X is one over 25 square? You&#x27;re over 25. Give me your goodbye. Let&#x27;s give a trick question. Either, actually, zero or, um, the other factor could be hero fell won over 25 squared. True between political hero. Now, this equation, you&#x27;re not going to have any solution when we see that. Is that the problem? A function here would be a problem with the shifted upwards above the X axis. And another way, T that is. We tried talking for actual end up taking the square root of a negative number. So no critical time we hear hear their only critical number, actually, zero and the only critical number. It must be the the minimum that we&#x27;re interested in finding X equals zero. Uh, when dysfunction of minimized. Remember the X original iwas uh, represented X coordinate of a point on this problem on the curb, and we want to find the point undercurrent, which, uh, have you know I&#x27;m different, though. What? Let&#x27;s finish finding a coordinate. Do you think the question here? So the y coordinate? Why would one over 10 times zero you did your affair? The coordinates are 00 This is the point on the function, which is close to the point. There come a point back. That&#x27;s just the origin.

Problem

Find the point on the ellipse $x=2 \cos t, y=\sin…

Problem 49 Hard Difficulty

Find the point on the parabola $x=t, y=t^{2},-\infty<t<\infty$
closest to the point $(2,1 / 2) .$ (Hint: Minimize the square of the
distance as a function of $t .$ )

Answer

$(1,1)$ is the point on parabola which is closest to $\left(2, \frac{1}{2}\right)$

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$(1,1)$ is the point on parabola which is closest to $\left(2, \frac{1}{2}\right)$

Thomas Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Michael J.

Integration Review - Intro

Definite vs Indefinite

George B. Thomas Jr.Thomas Calculus

Anna Marie V.

Kayleah T.

Caleb E.

Michael J.

George B. Thomas Jr.

Thomas Calculus