A question on domain

[fred2028]

For an optimization problem, such as what's the max area I can cover with x amount of fence, should I include 0 in the domain? According to http://tutorial.math.lamar.edu/Classes/ ... ation.aspx,

Note that the endpoints of the interval won’t make any sense from a physical standpoint if we actually want to enclose some area because they would both give zero area. They do, however, give us a set of limits on y and so the Extreme Value Theorem tells us that we will have a maximum value of the area somewhere between the two endpoints. Having these limits will also mean that we can use the process we discussed in the Finding Absolute Extrema section earlier in the chapter to find the maximum value of the area.

They include the endpoints, making it a closed interval. I guess essentially they mean that if the width is 0, then length must be infinity. However, when I graph some of these functions, a vertical asymptote exists at x=0, so I can't include it in my domain. When am I supposed to include it in my domain?

 

[Deleted member 4993]

For an optimization problem, such as what's the max area I can cover with x amount of fence, should I include 0 in the domain? According to http://tutorial.math.lamar.edu/Classes/ ... ation.aspx,

Note that the endpoints of the interval won’t make any sense from a physical standpoint if we actually want to enclose some area because they would both give zero area. They do, however, give us a set of limits on y and so the Extreme Value Theorem tells us that we will have a maximum value of the area somewhere between the two endpoints. Having these limits will also mean that we can use the process we discussed in the Finding Absolute Extrema section earlier in the chapter to find the maximum value of the area.

They include the endpoints, making it a closed interval. I guess essentially they mean that if the width is 0, then length must be infinity. However, when I graph some of these functions, a vertical asymptote exists at x=0, so I can't include it in my domain. When am I supposed to include it in my domain?

For optimization - you are looking for global min/max - within a given range.

A point with vertical asymptote will not be included in the range - if the problem is well-posed. In other words, in problems with physical realities - unbounded y (asymptotic behavior) will be excluded.

 

[fred2028]

For an optimization problem, such as what's the max area I can cover with x amount of fence, should I include 0 in the domain? According to http://tutorial.math.lamar.edu/Classes/ ... ation.aspx,

Note that the endpoints of the interval won’t make any sense from a physical standpoint if we actually want to enclose some area because they would both give zero area. They do, however, give us a set of limits on y and so the Extreme Value Theorem tells us that we will have a maximum value of the area somewhere between the two endpoints. Having these limits will also mean that we can use the process we discussed in the Finding Absolute Extrema section earlier in the chapter to find the maximum value of the area.

They include the endpoints, making it a closed interval. I guess essentially they mean that if the width is 0, then length must be infinity. However, when I graph some of these functions, a vertical asymptote exists at x=0, so I can't include it in my domain. When am I supposed to include it in my domain?

For optimization - you are looking for global min/max - within a given range.

A point with vertical asymptote will not be included in the range - if the problem is well-posed. In other words, in problems with physical realities - unbounded y (asymptotic behavior) will be excluded.

OK, but global extrema do not exist on open intervals. In the case of say 0 < x < 10, would 1 just find the highest/lowest of the local extrema instead of looking for a global?

 

[tkhunny]

global extrema do not exist on open intervals

Since when? Consider f(x) = -x^2 + 2x on [0,2]. Is the Global Maximum ANY different on (0,2)?

 

[fred2028]

global extrema do not exist on open intervals

Since when? Consider f(x) = -x^2 + 2x on [0,2]. Is the Global Maximum ANY different on (0,2)?

Oh shoot sorry you're right. But what if the graph which models the optimization question is not a polynomial of even degree? Then how would 1 find the global extrema?

 

[tkhunny]

There is a very big difference between asymtotic behavior such that the asymptote is not in the domain and arbitrary exclusion of part of the unrestricted Domain.

Consider f(x) = 1/(x-1) on [0,1] - This should make no sense, since x = 1 is not in the Domain.

Consider f(x) = 1/(x-1) on [0,1) - This should make sense, but you will have a hard time find a Global Maximum. It is unbounded.

Consider f(x) = x on [0,big) - This should make sense. Again, you will have a hard time find a Global Maximum. It is unbounded.

Consider f(x) = x on [0,1) - This should make sense, but you will have a hard time find a Global Maximum, even though it is bounded. This is the problem you suggested, since x = 1 is not in the Domain. It may be time to look up the "Supremum". The "Sup" doesn't have to be in the set.

Consider f(x) = x on [0,1] - This should make sense, and you should have no trouble finding a Global Maximum, but calculus will not find it for you. The derivative does not know you have chopped off the graph at x = 1. This is not a problem. You just have to check yourself.

Are we getting anywhere?