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Complete the solution to Exercise $8 .$ Make sure to justify your conclusions.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

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.

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Fill in the blanks.$$<…

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Fill in the blank(s) to co…

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Fill in the blank to make …

01:23

Fill in the blanks to comp…

01:58

00:53

Write a justification for …

00:36

Fill in the blanks.The…

01:20

Write expressions that giv…

00:42

Each calculation below is …

they asked us to actually follow through with this problem, they set up in the chapter where we have a factory here and a power plant power plant here and there's a river here, that's five miles wide. And this power plant um In the problem, well, in the second problem, it's one mile inland from the edge of the river. And so they tell us that the the length, the cost of cost to um put wire along the along the overland. Um Let's see. That was the the cost of quote um is 50 mile $50 per mile. So to go overland. So this cost $15 per mile and this costs $80 per mile. So they want to figure out where um where we should put this kind of a Transfer point here, I guess you could say um you know, and we know this is 2020 miles here. So where do we put this point here? Uh so then we can run cable here, I guess. They said uh under the water. So, well the coal cost is 50 times this plus 80 50 times this length plus 80 times this length. So this is our total cost. And again, I've used the constraint that you know that this is saying this is why X plus Y equals 20. So I've already kind of used that. So we um take the take the derivative of this respect. X. That actually put X one by seven equals zero. And we get this this ugly, ugly looking expression here now, what we can do with this is move on to the other side. Cross multiplied by the square root terms. And then um then square both sides. And in the end we wind up with this expression here. Um So we can see that it's 1/4 order polynomial. Fourth order terms over here and fourth order terms. Uh Yeah, experts here and experts here. So I mean I could um I didn't write this out as a polynomial. I think I did in my notes though. I should see if I have that. Oh yeah, it's well, first, yeah, yeah, you can multiply it out and figured out it's kind of got big big coefficients. Um So let's see here now, you know, normally you just can't, you know, this is the fourth order polynomial. But technically you can find a closed form solution to that. And if you use like computer algebra system, mathematical something, it will actually find a closed form solutions for you in terms of square roots and cube roots and forced roots and all kinds of things. Um and it will be so big and ugly that it will have absolutely no meaning. So the other thing is to do is just find it numerically. Um And you can just basically if you plot this thing, it kind of looks like uh this So there's some zero here, there's some some place where this equals this or if we move everything to one side that the difference of these is zero And that point turns out to be very, very close to 16. So we want to go 16.01 miles over here. So obviously this is this is going to look more like like this which really expect right because it costs much less to go overland. So we want to do overland pretty much as much as possible. But then straight here now, this is not the solution though, because again, it's just um it would cost more because we have to we have a lot more. We have a lot of overland and then the cost to make this a little bit less over water isn't very much because the distance this distance and this distance are not that much different. So in the end We go about 16 miles down river, or upriver from the power plant and then start um crossed. Okay?

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