Analysis and Calculus

Can someone give me an intuition on why the definite integral represents area under a curve ? Thanks .

I was trying to think if there is some rule for various 3-D shapes, like the interior angles of a triangle always total 180 degrees and it goes up by 180 for every side you add of a regular polygon, but I was wondering recently if there's some kind of property for 3-D shapes. Obviously 3-D shapes have more than one angle occurring at a single point, but if we have say, a triangular pyramid, is there something about the total radian measure of interior sphere arcs that totals up to like, 2pi radians something like that? From forming spheres. Like if you have a cube, and you draw spheres with centers at each vertices of each vertex of the cube, all of those spheres will al…

Hello! I'm trying to understand the following formula, and pretty much need help with all parts of it: If anyone out there could help me I'd be extremely happy ! Best regards, The Interpretator

I got to thinking about this from two recent threads on the nature of pi. We're all familiar with scribing a semi-circle using two points (A and C) and a right triangle, shown below. How then would the distance along the circumference of the semi-circle be computed using integration? Can it be done without using trigonometry, or is trig inherently part of the process? Pi would then be the ratio of the semi-circle's circumference to ½ of AC. I apologize for not thinking this out myself, but I haven't focused in this mathematical direction in quite some time.

y=(1/4)x^2; y=5-x^2 Find the volume of the solid obtained by rotating the region bounded by the these curves about x=7. Okay, I thought I needed to change the given equations to x= sqrt(4y) and x= sqrt(5-y), but I'm sure that's not right, because then there is no enclosed region. Now I have no idea how to do this one. Can anyone help me with this? Thanks.

I'm solving a set of equations as part of a simple fixed point optimization algorithm that I am implementing in MATLAB. I have a set of N numbers A={a(1),a(2),.......,a(N)}. Each a(i) is a member of the positive reals. No order or pattern is presumed in the values of A. I am trying to solve the following equation: sum(i=1:N){1/(a(i)+x)}=C To give a simple numerical example with N=4, A={1,3,2,0.5), and C=10: 1/(1+X)+1/(3+X)+1/(2+X)+1/(0.5+X)=10 I can solve x by multiplying through the denominator, creating a 4th order polynomial and then solving this polynomial, but this is a lot of work and not feasible for large datasets (e.g, I have a dataset with n…

link removed by moderator A new general and unifying arithmetical concept: The Rational Mean, which allows to generate, among many other new algorithms, those celebrated Lucas’s, Bernoulli’s, Newton’s, Halley’s, Householder’s root-approximating methods which up to now were considered the exclusive achievement of Infinitesimal Calculus. No derivatives, no trial-&error checkings, no geometry, no cartesian system, but just Simplest Arithmetic. These new high-order methods also embrace complex roots and the general algebraic equation, and from the solid evidence at hand, these arithmetical methods have no precedents in the mathematics literature. Indeed, it is reall…

Hello, I need to find a two-arguments function u(x,y) which satisfies six constraints on its derivatives. 1&2: On the first derivatives: du/dx>0 for all x & du/dy>0 for all y (so u is increasing in x and y) 3&4: On the second derivatives: d²u/dx²<0 for all x & d²u/dy²<0 for all y (so u is concave in x and y) 5&6: On the crossed derivatives: d²u/dxdy<0 for all x+y<theta (or at least y<theta) & d²u/dxdy>0 for all x+y>theta (or at least y>theta) (theta is a threshold) I found one specific function that satisfies those conditions: u(x,y)=xy+1-exp(theta-x-y) But I don't think this is the only …

Hi All, I hope the following book could be of some interest to you: link removed From the evidence at hand, these new high-order methods have no precedents in the math literature. Regards, Domingo Gomez Morin

If on a test or experiment you result above the average of the sample group that is generally regarded as good. However, what if you score above average, but the average is failry low...say out of a sample size of 1000, the average score is 40%, and your "above average" score is 49%, with the highest score being 59 (standard deviation is 10.7). Is this "above average" score really that great or is it just a little bit better? (these are not actual sample scores, just random numbers)

An equation of the following form is given: [latex]c=-a_1*\sin(\alpha)+a_2*\cos(\alpha)[/latex] next by expanding the equation above the following equation is obtained: [latex]c=-b_1*\sin(\beta-\delta)\gamma-b_2(\frac{\partial u}{\partial v})\dot \gamma[/latex] i have rewritten the first equation to: [latex]c=\sqrt{c_1^2+c_1^2}\cos(\alpha-\epsilon)=\sqrt{(c_1^2+c_2^2)(1-\sin^2(\alpha-\epsilon))}[/latex] where [latex]\epsilon=\tan^{-1}(c_2/c_1)[/latex] However, I don't think this is the correct path to obtaining the expanded equation and if it is I can't see what the next step should be. How do they obtain the expanded equation?

Hi, I've been working through some derivations for constitutive models of viscoelasticity, and they've been leading me to forms involving the Lambert (product log) function. The solutions get complex very quickly. In an attempt to simplify, I'm trying the simplest possible constitutive model having the form y=x*exp(dx/dt*t). Now this is where things get strange: Mathematica (Wolfram Alpha) returns a solution of y=x for Solve[y==x*E^[D[x,t]*t],x]. Any thoughts? Best wishes! David Kazmer Just following up a bit, an equivalent form is Solve[log(y/x)=t*dx/dt,x] which is recognized as a non-linear ODE and yields y Ei(-log(y/(x(t))))+log(t) = …

Hi all, I'm a new user , I'm Italian so sorry for my bad english. My problem is that : I have this second order system non linear: [latex]x(k+1) = a \cdot x(k) + b \cdot \sqrt{(x(k-1))} + c \cdot u(k)[/latex] I would like to have a new system that is the linearizazion of my one. I know how to linearize a first order system, but in this case how can i do ? Thanks

So if you have experience with calculus you know that you can use limits to sort of cheat physical reality and add an infinite number of infinitessimally small things to get a solid thing. However, I started thinking about compounding it. You can make a cube by adding an infinite number of infinitesimal squares, but what about making square? I postulate that there is a mathematical way to add up an infinite number of points to create lines, then have the spacing between those lines go to 0 to create a plane or square, and then have thickness and spacing of those squares go to 0 to create a solid cube all in one equation. Anyone have ideas? Or is it impossible?

OK.. SO I know basic math and SOME algebra, but it has never been my subject of expertise... Anyways... I use a card swiper and I get charged 2.75% of the price + tax/tip... so I am wondering how I would figure out, or what % tax I need to add to counter this 2.75% charge I get. I know charging 2.75 will NOT work because if I charge 2.75 that makes a 20 dollar charge would make it like 20.55 but... then I get charged the 2.75% which means I get charged roughly 0.56.. which means I only get 19.99.. which really isn't that bad.. but is there a percentage that will counter this to keep what I get right at 20.. or any dollar amount... so say if I charged 90 dollars for s…

If I say 1/0, how do I proved it's "undefined" and not some actual number? I can't use "undefined" in algebra to solve for a variable, why should it be a real solution? I could see how it makes sense orally, but it seems like there's weird unpredictable guidelines for where human assumption comes in and what the math carries out to be. Like if I say "x approaches infinity", that's used in all sorts of algebra but that's not an actual numeric thing that's just us basically saying "as x increases indefinitely", but in mathematics x never actually has a value of infinity. But anyway, if I say 1/0=undefined, and I have x = 0 * undefined, shouldn't I logically be able to …

What's the difference between convolution and crosscorrelation? I read the answer below, but I don't know enough math to understand it. Could someone clarify it for me, please? "The meaning is quite different. To see why in a simple setting, consider $X$ and $Y$ independent integer valued random variables with respective distributions $p=(p_n)_n$ and $q=(q_n)_n$. The convolution $p\ast q$ is the distribution $s=(s_n)_n$ defined by $s_n=\sum\limits_kp_kq_{n-k}=P[X+Y=n]$ for every $n$. Thus, $p\ast q$ is the distribution of $X+Y$. The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by $c_n=\sum\limits_kp_kq_{n+k}=P[Y-X=n]$ for every…

Probably a duh question: If we define a mathematical object made up of two identical forever-rotating circles, everything perfectly regular (circular circles, axes central, rates of rotation unvarying, etc.) with the rates of rotation set in a non-integral ratio, such as 1 to pi, that is, the circles have a ratio of rotation that is a non-terminating, non-repeating fraction, will any particular state of that object ever recur--will two points that pass each other on a line through both axes ever be in that position again? I think that, because the ratio of rotation is non-integral, no state of the system is ever going to be repeated. Of course, I am thinking in a plan…

For the following proof of the intermediate value theorem ,which i found in wikipedia: Proof: Let S be the set of all x in [a, b] such that f(x) ≤ u. Then S is non-empty since a is an element of S, and S is bounded above by b. Hence, by completeness, the supremum c = sup S exists. That is, c is the lowest number that is greater than or equal to every member of S. We claim that f© = u. Suppose first that f© > u, then f© − u > 0. Since f is continuous, there is a δ > 0 such that | f(x) − f© | < ε whenever | x − c | < δ. Pick ε = f© − u, then | f(x) − f© | < f© − u. But then, f(x) > f© − (f© − u) = u whenever | x − c | < δ (that is, f(x) > u for …

The Twin Primes Conjecture Sometime ago, when my hasty first attempt went wrong, I promised to come back sometime and give my ‘proof’ of the twin primes conjecture. As was rightly pointed out in the initial discussion it is not a proof as a mathematician would use the word, so I’ll call it an heuristic argument. Or, if you like, it's just a thought. I make no claims for it other than that I think it correct as far as it goes. All that matters to me is whether I have made mistakes. I’ll try to state it as briefly as possible. It is painfully simple. Please excuse the clumsy termimology. I have tried to be clear but clarity is not one of my talents. . As I make u…

think of taking a loan this way loan+loan-loan+loan=? or 100eu+100eu-100eu+100eu=?

the problem says: "show that the zero solution is nonlinear stable. For this, find the change of variable that transforms this system in a linear system" [math] \frac{dx}{dt}=-x + \beta (x^2+ y^2) [/math] [math] \frac{dy}{dt}=-2y + \gamma x y [/math] i tried with the method of eigenvalues of Jacobian matrix, and both eigenvalues are negatives, but my teacher says that this method is incorrect. Help please

Heydi ho everyone, I was doing some differentiation of quaternions, and if I have a quaternion, I can translate a point, e.g. pos = q t q* where t = q(0,x,y,z) simple, yes? If I have the exponent of the quaternion, the axis-angle (w). so my quaternions are q(w) and q(w)*. What would be the change in position with change in axis-angle (i.e., the exponential). d( q(w) t q(w)* ) / dw = ?? If that makes sense? Thanx, Bob

find x? e1/x - x =0