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Find the point on the curve $y=\sqrt{x}$ closest to $(3 / 2,0)$.

$$(1,1)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

09:17

Find the point on the curv…

01:27

03:27

02:32

Find all points on the cur…

03:23

this problem we have to find a point on the car vehicles squared of X that's closest to um See here we have zero. All right, so I just I changed this two X equals y squared. So we get rid of the square root but then we got to make sure that we count positive values for why. Um So here's our distance squared and we're going to minimize the distance squared because that's the same as minimizing the distance. So we have um substituting this into here. We get this and we can take the derivative of disrespect of Y and we get two times y minus two X. Not Y plus two Y q minus why not? So that is actually a cubic equation which means that generally speaking we're not going to be able to solve it. Um Well we can solve it in close form but the Russians are really ugly. There is a formula for solving cubic equations, like there is for quadratic equations but it's ugly. However, if um why not zero and X that is three. Have we plug that into here and we get the squared crime is to y times the quantity uh Y squared minus one. So that is easy to solve for zero and we get why one optimal solutions are either one or either zero or plus or minus one. Well, we've got to get rid of the minus one because we only have positive values for y. So we either have one zero or one and if if um x is if y is zero the next zero. So that's this point here. If x, y is one, the next is also one. That's this point here. So we can check the distance, here's our point and we can check the distance from here to here is um well 34 we have obviously, and the distance from here to here is 4 to 5/2, which is less. So this is the closest point. Um This is really kind of a not not a max more men. The minimum is obviously offered infinity, basically a saddle point and uh solution. So here it is, and we can see again that the line connecting these points is orthogonal to the tangent line on of this curve here. So that's basically we should always that should always be the case.

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