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Show that the solution to Example 5 may be found as follows: reflect $A$ to the other side of the river to $A^{\prime}$ as shown in Figure $14 .$ Draw a line connecting $A^{\prime}$ to $B .$ Where this line intersects the horizontal line is the required point $P$ the location of the pumping station. (Hint: Assume that some other point gives the minimum length. Deduce a contradiction by using the fact that the sum of the lengths of two sides of a triangle exceeds the length of the third side.)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

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University of Michigan - Ann Arbor

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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.

08:05

Solve the given problems.<…

06:19

Solve the problem in Examp…

01:12

On the same side of a stra…

we have to kind of gives a geometric approach to show from an example problem chapter. So the example problem, they had two cities, I guess here and here and there was a pumping plant or something here on the river and they wanted to know where it should be placed. So that we minimize the amount of pipe going. You can pipe, there are street city. Um so they suggest that one way we could do this geometrically is to say, okay, well we take this point and flip it to the other side of the river and we're assuming the river has no has no whip otherwise. Well, at least that's the pumping station was in the middle of the river. Otherwise we have a bit of a it would be not, there will be uh it wouldn't be, it wouldn't be valid because we have some with at the river and flip it. So the river is is infinitely or negative is negligible compared to these distances. And so they say that, well, we can show that the position of the pumping plant that optimizes that minimizes the length of pipe is actually just the one where the straight line between this city and this city crosses the river. Right? So if this was on this side, and so we can see that again, if it's on the other side. Well, that's the same problem, right? Because this length is equal to that length. And so they asked us to show that by saying, well, let's assume that we put the pumping plant somewhere else. But first of all, you can see that, you know the minimum, the shortest distance between two points is a line. So that should be this, so that's a total quite. But you can show that, you know why that is because um you take some other point here and then now you have this, well, you know, from the uh what is it, some kind of uh what is it the um you think my pythagoras? No? Uh .2. So yeah. Alright, fact that some of those, thanks of two sides of a triangle, the seats, the length of the third side. So, you know that that's just, you know, just a theorem from geometry that some of any two sides of the triangle, that is not just a line. Um The two sides, are you no longer than harder than any of the other? So that basically says that this, plus this has to be greater than that. And so no matter where we put this pumping station, we're going to have, you know, sides of a triangle that are going to be longer than this one. So this is going to be the minimum right here. Where basically uh we basically have just a straight line between a a prime mirror of the city and be

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