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Minimum Distance, find the points on the graph of the function that are closest to the given point.

$y=x^{2}-2,(0,-1)$

(ANSWER NOT AVAILABLE)

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 7

Optimization Problems

Limits

Derivatives

Differentiation

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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So you know it. We need to find a function where, Trevor, than the distance between point on a curve and the point they'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'm making simplify that to make and then you're going to be X squared. Uh, mining one squared. All right, You'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're going to make d squared are functions we escorted side of the equation. We'll get, uh, square X squared, minus one. Where and this we're going to make we're going to call it f of X now. So if we minimise at vexed, then for minimizing this B squared square, didn't But when the square of the beautiful minimize the This is awesome. In a mine. So that's okay. We can just minimize the simpler functions. And now we need upon the minimum the function. Uh, I'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're ready to find a minimum of dysfunction. Take the derivative. We'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'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'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're out and two X squared minus one equals zero. And here forget I equals zero. It's a critical point And here I saw for X and take the square root. So we'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's quite a first derivative. Uh huh. We're going to set up a number line for the value of driven it when we have euro. You'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'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're a fucking 10 into the derivative. Here. We'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'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'll find a derivative positive. You're and over here could plug in positive 0.1, and you'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've drawn it wrong. Thanks, man. I'm saying the correct word, but I'm sure the wrong picture. So from negative to positive, I mean, their function decreasing angry thing like this off. It'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'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't square. That means for these acts value. This is between the curve and the point of being minimized. Now, we'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'll get, uh, to over four. Oh, they did more than think I'm actually, uh, when we No matter, we need a positive or the negative. We're going to get a positive. The fourth minus two, which turned out to be negative three house. That's going to be this. Whether or not we're talking a positive printer, beauty or negative, it over to that number is being squared. All right, so let'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

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