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(a) John is at $B$, on a straight beach, 10 miles from $A$. Mary is in a boat in the sea at $C, 4$ miles from the beach (see Figure 12 ). Assuming Mary can row at 3 miles per hour and jog at 5 miles per hour, where along the beach should she land so that she may get to John in the least amount of time? Solve the problem if John is (b) 100 miles; (c) 1 mile from $A$.

(a) and(b) 3 miles(c) 1 mile from $A$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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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requires a little bit of thought. Um And I think they're kind of leaving you there with the with the question, So we have somebody in a boat out here and then another person on the beach here. The distance. So this perpendicular line from the boat to the distance from the beach to the boat, the minimum distance is A. And then the distance from there to be to this other person is B. So this is a, this is B. Now we're told that the person can row the boat at five mph and jog on the beach uh throw the boat at three miles an hour and jog on the beach or five miles. So what we want to know what is the optimal landing point for the boat? Now we need to, so I define this is C. So the distance from the perpendicular line from the original position of the boat to the beach throughout the afternoon point where they land. And so then this distance is square root of a squared plus C squared. And then this distance is, well technically I guess I should put an absolute value in here too is um be absolute value of B minus C. Because B could be less than C. So C. Is down here of B. Is here. Yet. You know, basically we could have a situation where we overshoot um So we need to be careful with that. So we can Right this. So a total time is the distance growing divided by three MPH Plus distance jogging divided by five mph. Now one way you can take through this absolute value of B is to do this. So that's basically defined as the that's the absolute value. So you can do that and you can use that to take the derivatives if you want. Or you can see that this winds up being just minus 1/5 times a sign, the SGN of B minus A or b minus C. And so you'll see here that you'll get like a B minus C. All over a square root of B minus c squared. So you know this is the absolute value and this is the value. So we just get this would be the sign of the argument. So we get this and now what we can do is in this first case they say, okay that's that A equals Now be was 10 miles And a. Was a. is four miles. So we Put that into here and we try to solve this for zero. Make this is equal to zero And we get an optimal value of four. A over three. A over four. But that's only true if B. is greater than three. Um and we'll see that in that's why in a second. So if for 10, see what, see would just be if S four C. Would be three miles. So Go you know here this would be five miles And this would also be five miles. Now if um if she is A if B is 100 in the second part of the problem then we can see that. Well we still get, we get um five miles and then uh This is three, this was three, so this was his five and this is seven in the first case, five and 7 And then we get five and 97. It was 100 miles. So it doesn't, If B is greater than three then we basically always row the boat on this diagonal here. A role that about five miles. Now if B is less than three Then we can see that if we just play around a little bit. So if we pull again a is four and B is you know something less than three we can see that we have no solution. There's no value with this equals zero. Um and that that's where basically we get into a problem so we have no value. What we can't make, this equals zero. But what we can see here is if we plot t. So say say this was equal to this was the case when c equals one. Okay. And this is T. And this is um mm no, this was the case. That insight, This is B or one and this is seat. So if we plot that we can see what happens is that there's a kink here. Now for other values of B we get something like this. Now get something like this, we still have a kink but it would basically, we'd have a king. So we have something like this, we have a minimum, that was not not at the king. So here we see that we have a kinky. So it's a discontinuous function here. I mean it's not a smooth function here. So the the derivative is not defined there, so we can see the driver that was negative here and positive here. So our minimum occurs at the kink and that's why we have to be careful here because that kink always occurs when C equals B. And so when C is when B is less than three, our minimum occurs at the kink here. And so this over here is a plot of this is T. And then this is, let's see this is B. And this is C. So when we, when we plot this like this, we can see that our minimum goes from here, the heat goes up like this be, we'll see you in this region the optimal. And then we go up here basically just C equals equals. Again, this is these are all with these are all with a equals four. And so we can see that then over here we just wind up with C was three For any be greater than three. So again, um you know, there's always this kink here, like if we take any value of being, but we wind up with a minimum away from the king. Um when B is greater than three, but we don't we don't have a minimum away from the king. We just have, the king could be is less than or equal to three. So again, um we have to be a little careful. So in the end, when he is when C. One B. Is one mile, then see also should be one mile. So she should just basically wrote directly to him.

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