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I got
but Mathematica tells me it's
I found these to be the same function by graphing. I understand how integrals can be in different forms and are still equal. This is true in many trigonometric integrals, where two functions may look entirely different, but are found to be equal or to differ by a constant when some identities are used to simplify. How are these equivalent? I couldn't find a way to simplify either one to make it look like the other expression. And also, how did mathematica arrive at a simpler answer? -- Yanwen ( talk) 02:19, 2 June 2009 (UTC)
what is the babylonian quadratic equation, used to solve simultaneous equations? what was their logic to solve quadratic equations in their times? —Preceding unsigned comment added by Thee madhav ( talk • contribs) 06:54, 2 June 2009 (UTC)
Conic section says one of the conics is an ellipse (not an oval). I know oblique section of a cylinder is an ellipse. Thinking from there, the section of a cone seems to be an oval, because a cone is in a shape of a cylinder augmenting to the bottom (If one end of an ellipse is augmented, it is likely to be an oval. I tried cutting a cone made with paper, and that resulted in subtle, but like a tear drop shape (have I done it wrong?) Like sushi ( talk) 10:53, 2 June 2009 (UTC)
Dandelin spheres!
If you can't understand the algebraic equations given in response to your question about ellipses, then do look at the article titled Dandelin spheres. Michael Hardy ( talk) 15:28, 2 June 2009 (UTC)
The concept of ellipse is often defined by saying it's the set of points for which the sum of the distances from two fixed points is a fixed constant. You're saying you don't know why that's an ellipse? You must be using some other definition. What is it? Michael Hardy ( talk) 06:36, 3 June 2009 (UTC)
Hi, I'm looking for a good textbook on solid geometry, preferably with problems and answers. I'd like it to cover stuff like proving that the shortest surface distance between any two points on a sphere lies along a great circle between them. Also conic sections and intersections between, say, planes and various solids. Any suggestions welcome, thanks, It's been emotional ( talk) 13:51, 2 June 2009 (UTC)
Engaged in a seminar about functional anlysis, we came across a problem. Let us consider the space which is the set of all real continuously differentiable functions on the interval [a,b] given. And define the functional
on this space. So the functional maps a function x to its derivative evaluated at the midpoint of the interval. On this set, define the norm to be
and define the norm of a functional on this space to be
Now basically I am trying to determine the norm of my above defined functional f. What I have done is that I have shown that this norm is bounded, meaning
so the constant is one in this upper bound. Now I know that the norm of f is less than or equal to one. My guess is that the norm of f is exactly one but how can I show that the norm of f is exactly equal to one? If it is not exactly one, then what is it and how can I find it? Is there a lower bound I can put on the norm at least? Can I say anything about the norm of f? Can the norm even be exactly determined? Thanks!
-Looking for Wisdom and Insight! (
talk) 23:26, 2 June 2009 (UTC)
That was amazing. That is exactly what I needed. I got it. Thanks! -Looking for Wisdom and Insight! ( talk) 04:10, 3 June 2009 (UTC)
As part of some math I am trying to figure out, I need to show that for some small () for ; and I am having trouble figuring it out. I feel like there is some famous inequality I should use here. I know that the right side is the first two terms of the binomial expansion of the left side, but that is only a convincing argument if is positive; when is negative, I know that the inequality is still true (from graphing it), but the binomial expansion argument isn't that simple anymore because the binomial expansion contains negative terms also, and so it is harder to say that the contribution from those terms will add up to a positive number. Other simple things I can do, like , don't work because the inequality is going the wrong way. I know from graphing it that I could argue that the left side is concave up (as a function of ), and that the right side is the first order Taylor polynomial at 0, and so has to be a lower bound, but that seems way too complicated. Thanks, -- 131.179.33.215 ( talk) 23:38, 2 June 2009 (UTC)
OK, let's try mathematical induction:
There you have it. Michael Hardy ( talk) 06:31, 3 June 2009 (UTC)
Alternate proof: Define and show that:
Combine (1)-(3) to get the result. Note that this proof does not rely on being an integer, and the proof technique works for many other inequalities, including . Abecedare ( talk) 07:18, 3 June 2009 (UTC)
Mathematics desk | ||
---|---|---|
< June 1 | << May | June | Jul >> | June 3 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
I got
but Mathematica tells me it's
I found these to be the same function by graphing. I understand how integrals can be in different forms and are still equal. This is true in many trigonometric integrals, where two functions may look entirely different, but are found to be equal or to differ by a constant when some identities are used to simplify. How are these equivalent? I couldn't find a way to simplify either one to make it look like the other expression. And also, how did mathematica arrive at a simpler answer? -- Yanwen ( talk) 02:19, 2 June 2009 (UTC)
what is the babylonian quadratic equation, used to solve simultaneous equations? what was their logic to solve quadratic equations in their times? —Preceding unsigned comment added by Thee madhav ( talk • contribs) 06:54, 2 June 2009 (UTC)
Conic section says one of the conics is an ellipse (not an oval). I know oblique section of a cylinder is an ellipse. Thinking from there, the section of a cone seems to be an oval, because a cone is in a shape of a cylinder augmenting to the bottom (If one end of an ellipse is augmented, it is likely to be an oval. I tried cutting a cone made with paper, and that resulted in subtle, but like a tear drop shape (have I done it wrong?) Like sushi ( talk) 10:53, 2 June 2009 (UTC)
Dandelin spheres!
If you can't understand the algebraic equations given in response to your question about ellipses, then do look at the article titled Dandelin spheres. Michael Hardy ( talk) 15:28, 2 June 2009 (UTC)
The concept of ellipse is often defined by saying it's the set of points for which the sum of the distances from two fixed points is a fixed constant. You're saying you don't know why that's an ellipse? You must be using some other definition. What is it? Michael Hardy ( talk) 06:36, 3 June 2009 (UTC)
Hi, I'm looking for a good textbook on solid geometry, preferably with problems and answers. I'd like it to cover stuff like proving that the shortest surface distance between any two points on a sphere lies along a great circle between them. Also conic sections and intersections between, say, planes and various solids. Any suggestions welcome, thanks, It's been emotional ( talk) 13:51, 2 June 2009 (UTC)
Engaged in a seminar about functional anlysis, we came across a problem. Let us consider the space which is the set of all real continuously differentiable functions on the interval [a,b] given. And define the functional
on this space. So the functional maps a function x to its derivative evaluated at the midpoint of the interval. On this set, define the norm to be
and define the norm of a functional on this space to be
Now basically I am trying to determine the norm of my above defined functional f. What I have done is that I have shown that this norm is bounded, meaning
so the constant is one in this upper bound. Now I know that the norm of f is less than or equal to one. My guess is that the norm of f is exactly one but how can I show that the norm of f is exactly equal to one? If it is not exactly one, then what is it and how can I find it? Is there a lower bound I can put on the norm at least? Can I say anything about the norm of f? Can the norm even be exactly determined? Thanks!
-Looking for Wisdom and Insight! (
talk) 23:26, 2 June 2009 (UTC)
That was amazing. That is exactly what I needed. I got it. Thanks! -Looking for Wisdom and Insight! ( talk) 04:10, 3 June 2009 (UTC)
As part of some math I am trying to figure out, I need to show that for some small () for ; and I am having trouble figuring it out. I feel like there is some famous inequality I should use here. I know that the right side is the first two terms of the binomial expansion of the left side, but that is only a convincing argument if is positive; when is negative, I know that the inequality is still true (from graphing it), but the binomial expansion argument isn't that simple anymore because the binomial expansion contains negative terms also, and so it is harder to say that the contribution from those terms will add up to a positive number. Other simple things I can do, like , don't work because the inequality is going the wrong way. I know from graphing it that I could argue that the left side is concave up (as a function of ), and that the right side is the first order Taylor polynomial at 0, and so has to be a lower bound, but that seems way too complicated. Thanks, -- 131.179.33.215 ( talk) 23:38, 2 June 2009 (UTC)
OK, let's try mathematical induction:
There you have it. Michael Hardy ( talk) 06:31, 3 June 2009 (UTC)
Alternate proof: Define and show that:
Combine (1)-(3) to get the result. Note that this proof does not rely on being an integer, and the proof technique works for many other inequalities, including . Abecedare ( talk) 07:18, 3 June 2009 (UTC)