Analysis and Calculus

Hi Guys, I've been stuck with the proof of this particular statement, specifically the comments at the end of the statement. As follows: Prove that If a < b are real numbers then a < a + b/2 < b. (How do you know that 2 > 0#? What exactly is 2?) My attempt, as follows: since a < b, then 0 < b - a (by definition.) *assuming that we know what b/2 is, and that b/2 < 0 such that -(b/2) > 0 then, we know that: b - a - b/2 = b - (a + b/2) > 0 ----> b > a + b/2. NB:[-a - b/2 = -(a + b/2) can be justified] the a< a + b/2 ---> don't know how to go about proving this part! *Can this assumption be …

So I know the normal derivative and anti-derivative relationships, velocity, position, acceleration, but what do you get when you take the anti-derivative of a position tie graph? What does that represent? normally it goes m/s^2 -> m/2 -> m -> ? What does the "un-slope-ing" of a position time graph do?

anyone can explain me why

So I know most shapes have an inverse, some of which do and don't have defanite edges, I know that you can program a transformation of a shape to its inverse, but I don't get what you do exactly especially for 3-D shapes. If I have a square, the "inverse" of a square seems be some kind of infinitely stretched thing with an open hole in the middle with arc lengths whos endpoints correspond to the vertices of a square. How would I see what it looks like for say...an inverse of a tube?

So I know the common physics ax^2+bx+ or aT^2+VoT+X0 among its other various geometric uses, and I've seen a couple 4th degree polynomials in advanced particle physics, but where would you find something like X^9-2.3x^8-45x^5+23x^4+30 in nature? Or where would you find pseudo polynomials like x^(3/2)+1/2x^(1/2)-x^(-1/3)?

I'm reviewing a calculus test I took and ran into this problem, which I got right but don't remember how... Find the equation of the tangent line to the curve y = f(x) which is parallel to the line 3x - 4y = 1 y = 3/4x - 1/4 So my answer was y = 3/4x + 5/4, where did the 5/4 come from? Without points how can we find what b is? Or am I missing information from the problem you think? Thanks.

Hi all, I am researching a hypothesis and looking for anyone who is familiar with differential topology (specifically Einstein manifolds). I have access to the Besse book Einstein Manifolds but am also looking for any open questions in differential topology that I am not aware of. I am attempting to develop a solid proof link between the Laplacian and Einstein manifolds (listed in the book as not found yet). The math is pretty basic, sort of a gauge theory of the Laplacian. In graphical form (as the equations might be confusing without them) there is no distinction between the derivatives of the following plots: therefore for a higher dimension …

Does the Dirac Delta function have a residue? I feel like it should, considering the close parallels between delta function integral identities and theorems in complex analysis, namely the Cauchy integral formula and residue theory. I'm unsure exactly how they tie together (if they do).

Our teacher told us to solve this: e^x = -x We are not even close to this topic yet and he also said it was just a joke, but now I'm really curious how to solve this. Would someone be so nice to explain it to me? I know it is difficult, because not even my teacher has solved it yet... but I find this so fascinating that someone can actually find an answer xD

Ok I know what an irrational number is suppose to be, but how do we know if a number is actually irrational? It can't be expressed in a fraction that we know of, but there are infinite numbers, there could be a number that is so big we haven't counted to it yet, and not only that but we can't even generate them at will, in order to we'd have to spend an infinite amount of time writing out decimal places. So I know pi is suppose to be irrational, but how can it be proven that it isn't countable? I don't know if it has to do with cantor's proof or not, I don't fully understand it as I haven't taken set theory. Any explanations?

[latex]\displaystyle \int \frac{dx}{a\sin x + b\cos x + c\tan x} \hspace{10mm}a,b,c \in \mathbb{R} \ne 0[/latex] [latex]let \; t = \tan \left(\frac{x}{2}\right) \Rightarrow dt = \frac{1}{2} \sec^2 \left(\frac{x}{2}\right) dx[/latex] [latex]\Rightarrow dx = \frac{2\, dt}{\sec^2\left(\frac{x}{2}\right)} = \frac{2\, dt}{1+ t^2}[/latex] [latex]\sin \left(\frac{x}{2}\right) = \frac{t}{\sqrt{1 + t^2}}[/latex] [latex]\cos \left(\frac{x}{2}\right) = \frac{1}{\sqrt{1 + t^2}}[/latex] [latex]\Rightarrow \sin x = 2\sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) = \frac{2t}{1 + t^2}[/latex] [latex]\Rightarrow \cos x = \cos^2 \left(\frac{x}{2}\right…

Hi, For any odd composite 'N', let u = (N-1)/2, v = u+1, then u^2(mod p) = v^2(mod p) if and only if 'p' is a factor of 'N'. For more info advertising link removed

I was just messing around with integrals to see if I could derive a general solution of this integral: [latex]I = \displaystyle \int \frac{dx}{\sqrt{c+bx+ax^2}}[/latex] and I was wondering if anybody could confirm my result. [latex]c+bx+ax^2 \Rightarrow c + a\left(x^2 + \frac{b}{a}x\right)[/latex] [latex]c+a\left[\left(x+\frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right][/latex] [latex]\Rightarrow \left(c-\frac{b^2}{4a}\right)+a\left(x+\frac{b}{2a}\right)^2[/latex] [latex]\Rightarrow \displaystyle \int \frac{dx}{\sqrt{\left(c-\frac{b^2}{4a}\right)+a\left(x+\frac{b}{2a}\right)^2}}[/latex] [latex]\Rightarrow \frac{1}{\sqrt{c-\frac{b^2}{4a}}} \displa…

How to find the imporper integral from - 00 to 00 of f(x) = (e^(ikx)).(1-e^x / 1+e^x) where k is a real fixed number ?? I tried to write e^ikx = coskx + i sinkx but i don't know what to do with the term multiplied by e^ikx which is (1-e^x / 1+e^x) ?? Thank you

Dear All: Please throw some light and guide me on how I should take it forward. Please find the attached image & then read through the following. Constraints: 1. The minimum distance between the holes are 200mm 2. The maximum distance between the holes are 600mm 3. The distance between the end of the bar and the nearest hole should be at least 70mm 4. The mill bars should have at least 2 holes The standard length of the mill bar is a constant, say 1500mm But the length of the plate, as well as the position of the holes, are variables. Keeping all the above mentioned constraints, we need to accommodate the bars on the plate. You may cut the bars into any le…

Please, tell me is this the true: http://www.scribd.com/doc/118430547/GEOMETRY-WITHOUT-A-SECRET Thank you!

The logic that odd composite with least difference will be factored easily and large difference would factored hardly is wrong. B'coz whatever be the difference between the factors their exist Best Fermat Factors to make the Fermat factorization easier. Please follow the link to know more. <link removed by moderator>

Not sure if this the right place for the question. Please move this post if not. Is there anyone here able to help me understand how R's Z-function is used to create his famous landscape? I will never understand the Z-function but am happy to treat it as a black box. What I'm trying to grasp, among other things, is what numbers have to be used as inputs in order to produce the necessary outputs. My thought is that each output value must have an input value associated with it. Yet I've never seen a discussion of the inputs, only the outputs. So maybe I'm misunderstanding something very basic. Just a discussion in natural language would be fine. I am no…

Hi all, there's this function: y=(x^2-6*x+5)/(x-1) which can be reduced to y=(x-5)(x-1)/(x-1) and then to y=x-5 The question asked me to determine the domain and range of this function. Here's the problem, when the function is y=(x^2-6*x+5)/(x-1), the domain is every real number except 1. But when it's been reduced to y=x-5, the domain should be every real number. However, the answer stated that the domain is still every real number except 1. How should I solve this problem?

Hi all, I am currently learning Jacobian and I wonder where can we apply this thing in the engineering field?

When you find the Maclaurin series form of f(x)= e^(-3x), you will get the summation from n=0 to infinity of (-1)^n * ((3^n)/n!) * x^n. The question is, what is the reason for assuming that the X factor is always x^n, how do we know its not x^(2n) or x^(3n)?

I was having a looks at multiple integrals, line/surface/volume integrals and the like the other week, and decided to try some problems, but this one stumped me: [math] \int \int_S xz\mathbf{i} + x\mathbf{j} + y\mathbf{k}\: \textrm{d} S [/math] where S is the unit hemisphere of radius 9 for y >= 0 I thought I could change the variables to spherical co-ordinates, but I don't see how that would work with the particularly nasty stuff you'd get for the [math] \sqrt{\left( \frac{\partial z}{\partial x} \right) ^2 + \left( \frac{\partial z}{\partial y} \right) ^2 +1} [/math] along with the square roots necessary in writing z in terms of x and y. Basically t…

I already placed such subject on other site. But anybody can tell nothing! I suggest to discuss's theme excerpts from this site

"In 1924, Stefan Banach and Alfred Tarski*** proved that it is possible to cut one sphere into five pieces that can be recombined to give two spheres, each the size of the original. Take any two sets not extending to infinity and containing a solid sphere each; then it is always possible to dissect one into the other with a finite number of cuts. In particular it is possible to dissect a pea into the Earth, or vice versa"...Motion Mountain. What is the applicable implication of this? And does anyone has a link or blog that could show the mathematics that was used?