mm-1779 === Subject: Re: What is y=f(x) > I know the basic concept of y equals a function of x. But can someone > give a beeter explanation of this funciton? y is just a more compact notation for f(x) === Subject: Re: Test News - > test news able server > Escuse me > -- > Azucena > Fondos de Escritorio Eroticos > http://www.personal.able.es/ensoriano === Subject: rational zeros test for polynomial functions?Real World? This will seem a bit dumb, but I am curious. Do mathematical analysts in the real world use the rational zeros test and descartes rule for signs to work with polynomial functions, but outside of mere academic objectives, like to be truly used for realistic practical things? G C === Subject: need help with limits I need help in finding limits(or showing that a limit doesnÇt exist), preferably using epsilon-phi notation. example 1) lim (x^2+y^2) / y (x,y)--> (0,0) this limit doesnÇt exist, but how do i show it ? example 2 ) lim ( x^2*(y-1)^2 ) / ( x^2+(y-1)^2 ) (x,y)--> (0,1) I have NO idea how to atack this one. The solutions manual state that 0 <= abs [( x^2*(y-1)^2 ) / ( x^2+(y-1)^2 ) ]<= x^2 (and i donÇt see how this is true). And they work out the problem from there. In short. i DON`T see any general procedure for taking limits with complicated expressions and i donÇt see any general pattern in the way the solutions manual solve the problems. It seems completely arbitrary. I need someone to enlighten me on how to atack these kind of problems. === Subject: Re: need help with limits >I need help in finding limits(or showing that a limit doesnÇt exist), >preferably using epsilon-phi notation. >example 1) > lim (x^2+y^2) / y >(x,y)--> (0,0) >this limit doesnÇt exist, but how do i show it ? If you're going to ask for people to help you on elementary problems, at least have the integrity to give a valid e-mail address. The easiest and usual way to show it is to find two methods of approching (0,0) which yield different results. Try approaching along x = 0 and along y=x^2 >example 2 ) > lim ( x^2*(y-1)^2 ) / ( x^2+(y-1)^2 ) >(x,y)--> (0,1) >I have NO idea how to atack this one. The solutions manual state that 0 <= >abs [( x^2*(y-1)^2 ) / ( x^2+(y-1)^2 ) ]<= x^2 >(and i donÇt see how this is true). And they work out the problem from >there. >In short. i DON`T see any general procedure for taking limits with >complicated expressions and i donÇt see any general pattern in the way the >solutions manual solve the problems. It seems completely arbitrary. I need >someone to enlighten me on how to atack these kind of problems. === Subject: Re: need help with limits >I need help in finding limits(or showing that a limit doesnÇt exist), >preferably using epsilon-phi notation. >example 1) > lim (x^2+y^2) / y >(x,y)--> (0,0) >this limit doesnÇt exist, but how do i show it ? > If you're going to ask for people to help you on elementary problems, > at least have the integrity to give a valid e-mail address. Why, so the poor guy can get 3000 swen virus messages a day? === Subject: Re: need help with limits > If you're going to ask for people to help you on elementary problems, > at least have the integrity to give a valid e-mail address. > > Why, so the poor guy can get 3000 swen virus messages a day? fancy you rewrite your email, i.e. me at home dot com and similar. Computing power of today is big enough and the techniques are available to decode such simple manipulations. You might as well killfile, scoring or whichever technique you fancy. //M.F. -- Sigblock empty. By choice. === Subject: Re: need help with limits > lim (x^2+y^2) / y >(x,y)--> (0,0) >this limit doesnÇt exist, but how do i show it ? My analysis is really rusty, so this may be a very naive question, but _why_ doesn't the limit exist? The expression is equivalent to x*(x/y) + y, and I should think the limit is 0 even though the function is undefined at (0,0). -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com reduces the number of useful answers you get. http://www.cs.tut.fi/~jkorpela/usenet/laws.html === Subject: Re: need help with limits >> lim (x^2+y^2) / y >>(x,y)--> (0,0) >>this limit doesnÇt exist, but how do i show it ? >My analysis is really rusty, so this may be a very naive question, >but _why_ doesn't the limit exist? The expression is equivalent to >x*(x/y) + y, and I should think the limit is 0 even though the >function is undefined at (0,0). up): the problem is that the limit is different on different approach paths. -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com reduces the number of useful answers you get. http://www.cs.tut.fi/~jkorpela/usenet/laws.html === Subject: Re: need help with limits > lim (x^2+y^2) / y >(x,y)--> (0,0) >this limit doesnÇt exist, but how do i show it ? > My analysis is really rusty, so this may be a very naive question, > but _why_ doesn't the limit exist? The expression is equivalent to > x*(x/y) + y, and I should think the limit is 0 even though the > function is undefined at (0,0). Let (x,y) -> (0,0) along the route y = x^3. === Subject: Re: need help with limits >I need help in finding limits(or showing that a limit doesnÇt exist), >preferably using epsilon-phi notation. >example 1) > lim (x^2+y^2) / y >(x,y)--> (0,0) >this limit doesnÇt exist, but how do i show it ? In general the trick is to show that the expression has different limits as (x,y) approaches the origin along different paths. It may help to recognize that x^2 + y^2 is the square of the distance from the point to the origin. Call this distance r; then the function in question is r^2 / y. If you approach the origin along the y-axis, r = y, r^2 / y = y, and the limit is clearly 0. Can you approach the origin in such a way that r^2 / y tends to some other limit? Well, r^2 / y = y + x^2 / y, where the first term will evidently approach 0, so you'll need to make the second term increase. This means that you'll need a path on which y gets small a *lot* faster than x. How about a circular arc tangent to the x-axis at the origin, say the lower half of (y - 1)^2 + x^2 = 1? Along this path we have y^2 - 2y + 1 + x^2 = 1, or r^2 = 2y, whence r^2 / y = 2. You could also have tried a simple parabolic path, y = x^2: along this path (x^2 + y^2) / y = (x^2 + x^4) / x^2 = 1 + x^2, which approaches 1 as (x,y) approaches the origin along this path. >example 2 ) > lim ( x^2*(y-1)^2 ) / ( x^2+(y-1)^2 ) >(x,y)--> (0,1) >I have NO idea how to atack this one. The solutions manual state that 0 <= >abs [( x^2*(y-1)^2 ) / ( x^2+(y-1)^2 ) ]<= x^2 >(and i donÇt see how this is true). The first inequality is of course true by definition of absolute value, so I expect that you're having trouble with the second. Let a = x^2, b = (y-1)^2; then you're looking at |ab / (a + b)|, where a and b are known to be non-negative. Since a and b are non-negative, you can forget about the absolute value signs: they're not doing anything. The claim is then that for non-negative a and b, ab / (a + b) <= a (presumably on the further assumption that a + b isn't zero). But this is clear: since a and b are non-negative, 0 <= b / (a + b) <= 1, and hence ab / (a + b) <= a. >And they work out the problem from there. By observing that if (x,y) --> (0,0), then necessarily x --> 0 and hence x^2 --> 0. Thus, the quotient in question is trapped between 0 and a function, x^2, that is approaching 0; by what is sometimes called the sandwich theorem, the quotient itself must approach 0. >In short. i DON`T see any general procedure for taking limits with >complicated expressions and i donÇt see any general pattern in the way the >solutions manual solve the problems. It seems completely arbitrary. I need >someone to enlighten me on how to atack these kind of problems. There really isn't a general procedure. The first step is to decide whether the limit is likely to exist or not, since this will determine what you do next; unfortunately, it takes some experience to achieve a high level of reliability. If you think that the limit does not exist, try the approach that I mentioned in connection with the first problem. If you think that it does exist, then you have to hunt around for some way to demonstrate it, and this will generally depend very much on the function. In your second problem the solution is achieved when you recognize that the fraction is bounded by something that is itself known to be approaching 0. Brian === Subject: Seating problem Hi I have been pulling my hairs out but still do not get the solution to the following seating problem, I would appreciate any help. The problem is: 4 identical twins (ie indistingusiable) are to be seated along one side of a dining table. In how many ways can they do so if no two twins are sitting together? I know the answer is 864, but I dunno how to get it. Please help! Anthony === Subject: Re: Seating problem > Hi > I have been pulling my hairs out but still do not get the solution to the > following seating problem, I would appreciate any help. > The problem is: > 4 identical twins (ie indistingusiable) are to be seated along one side of a > dining table. In how many ways can they do so if no two twins are sitting > together? > I know the answer is 864, but I dunno how to get it. Please help! > Anthony So, as I understand it, we want the number of arrangements of AABBCCDD so that no two consecutive letters are the same. The number of arrangements with no restrictions is 8!/2^4. Now let's use inclusion/exclusion to count the number of arrangements in which two consecutive numbers _are_ the same. The number with AA is 7!/2^3 so the number with at least two consecutive numbers is 4*7!/2^3. The number with AA and BB is 6!/2^2 so the number with at least two pairs of consecutive letters is 6*6!/2^2. The number with AA BB CC is 5!/2 so the total number with at least three pairs of consecutive letters is 4*5!/2. The number with AA BB CC DD is 4!. Putting it all together, the number with no consecutive numbers is 8!/2^4 - 4*7!/2^3 + 6*6!/2^2 - 4*5!/2 + 4! = 854. All of the above counting was done like the MISSISSIPPI problem if that helps. For example to count arrangements with AA we have [AA]BBCCDD. Seven objects total with B, C and D each being repeated twice. That gives 7!/(1!*2!*2!*2!) distinct arrangements. -- Paul Sperry Columbia, SC (USA)