Analysis and Calculus

Hey guys, based on 2nd post in the link [math]\frac{(1+h)^2 - 1}{h} = \frac{1 + 2h + h^2 - 1}{h} = \frac{2h + h^2}{h} = \frac{h(2 + h)}{h} = 2 + h[/math] we can cut and all of that if x = 1. but how about if we express the answer in terms of x? [math]\frac{(x+h)^2 - x}{h} = \frac{x^2 + 2xh + 2h - x}{h} = ?[/math] seems like quadratic equation, doesn't it will produce 2 answer? (where we only get one gradient(tangent) only, right?) btw, it's still not available to be answered because the h can't be 0.

Here's a program I wrote in Structured BASIC on my C64 computer: 100 REM COMPUTING PI USING TRAPEZOIDS 110 REM DOTS REPRESENT INDENTATIONS 120 CALL INIT 140 CALL MAIN 160 CALL OUTPUT 180 END 200 : 220 PROC INIT 240 .....TI$="000000" 260 .....A=0 280 .....B=1 300 .....X%=1 320 .....N=10 340 ENDPROC 360 : 380 PROC MAIN 400 .....DX=(B-A)/N 420 .....PRINT:PRINT "DX =";DX 440 .....DX=-DX 460 .....I=-1 480 .....PRINT:PRINT" I ";" X ";" Y ";" T":PRINT 500 .....LOOP 520 ..........I=I+1 540 ..........X=X+DX 560 ..........D=1-X*X 580 ..........REM 600 ..........IF D<0 620 ..............D=0 640 ..........ENDIF 660 ..........REM 680 ..........Y=2*…

how do we prove that : AεΑ is true or false ??

How do you prove that something proves something in number theory? Or how do you prove that something is proven?

Hi, I came across this theorem and decided to prove it, as follows: Theorem: A set [math]A \subseteq R[/math] is bounded if and only if it is bounded from above and below. I would like the prove the converse of the above statement; If a set [math]A \subseteq R[/math]is bounded from above and below, then it is bounded. Let [math]M = |M_{1}| + |M_{2}|[/math]and using this preliminary result I proved earlier [math]-|a| \leq a \leq |a|[/math]. Now, [math]\forall a \in A[/math] we have [math] a \leq M_{1}[/math] ---> definition of bounded from above. and [math] M_{2} \leq a[/math] ---> definition of bounded from below. Using the result: [math]-|a…

I'm trying to come up with a general formula to describe irrational numbers, by using the general summation of (1/n) , like an infinite summation of (1/3) = .33333333333 = 1/3, which isn't irrational, but I suspect irrational numbers can be decribed by some sort of summation that modulates with a base 10 or possibly a base = to some arithmetic form of n as to have a different modulus every time and thus never repeat. Though, every time I've ever thought of some cool mathematical thing, it terms out some dead scientist thought of it 100 years ago or more, so I'm just wondering if this has already been done.

Hello, I was wondering if someone could help me to interpret the following integral calculus problem: (Please open and study the attached file) Many thanksIntegral Problem.pdf

Basically I'm just wondering what happens if you use create a quadratic/cubic/quadic formula from scratch with the equation system substitution with the needed amount of data points but end up using a point that's not actually possible as part of the equation, like let's say I have just x^2, and for my data point I have 1,20 for the x,y substitution. What would I see? I would do it myself but creating polynomial formulas from scratch takes a long time for me, so I'm just wondering if anyone else has done than and can tell me what happens or what I should look for.

I've been trying to integrate this function by substitution, and it doesn't seem to be getting me to the correct place. I'm not sure I fully understand how to use substitution. [math] y = \int\sqrt{x^3 - 1} dx [/math] I've only ever dealt with substitutions where you will and up with [math] du = a dx [/math] where a is a constant, but if I make the substitution [math] u = x^3 - 1 [/math] then I end up with [math] du = 3x^2 dx [/math] And you can't just slap 3x2 back into the the integrand. Can someone integrate this and tell me what you need to do?

How to integrate f(x) = lnx on the circle (x-a)^2 + (y-b)^2 = R^2 ??? I will have ti use complex analysis Thank you

Hi all, I have power load data for a specified period (about 35 days) , I need to model it using neural networks and then fuzzy logic. for example for the first day: 9:02:01 183 9:22:14 198 9:42:27 195 10:02:40 198 10:22:53 203 10:43:06 190 11:03:19 184 11:23:32 219 11:43:45 227 12:03:58 232 12:24:11 258 12:44:24 218 13:04:37 213 13:24:50 211 13:45:03 215 14:05:16 233 14:25:29 223 14:45:42 220 15:05:55 232 15:26:08 224 15:46:21 233 16:06:34 236 16:26:47 223 16:47:00 223 17:07:13 223 17:27:26 237 17:47:39 241 18:07:52 262 18:28:05 253 18:48:18 255 19:08:31 253 19:28:44 242 19:48:57 250 20:09:10 243 20:29:23 248 20:49:36 236 21:09:49 227 21:30:02 236 21:50:15 228 22…

Hi, I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion. Here's what I have been trying to get at: [LATEX]\lim_{z\to z_o} z^2+c = {z_o}^2 +c[/LATEX] [LATEX] |z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta[/LATEX] [LATEX]LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|[/LATEX] [LATEX]=|z-z_o||\overline{z+z_o}|[/LATEX] [LATEX]=|z-z_o||\bar{z}+\bar{z_o}|[/LATEX] [LATEX]=|z\bar{z} +z\bar{z_o} -{z_o}\bar{z} -z_o\bar{z_o}|[/LATEX] [LATEX]=| |z|^2 -|z_o|^2 +2Im(zz_o) |[/LATEX] [LATEX]\leq||z|^2 -|z_o|^2 +2|z||z_o|| \ (because\ Im(z)\leq|z|)[/LA…

How to find the limit of the following complex function as z --> 0 f(z)= [ sin(tanz)- tan(sinz)] / [ arcsin(arctanz)- arctan(arcsinz)]

Hi, I am attempting on developing a basic mapping program ( very basic) an I am wondering if there is a way to approximate the shape of a function based on a few data points (3-5). I know that the function will be parabolic in nature and am wondering which numerical method or series I should begin looking at to guess at which point the maximum will be at. Background: I have 3-5 sensors (a set distance apart) that measure the time that an object is blocking the sensor. They return a value corresponding to the time that the sensor was blocked. For the sake of argument let’s say that the values I get are 2 seconds for sensor 1, 5 seconds for sensor 2 and 4 sec…

I was wondering what the rules are for a series to converge or not. In my calculus class our professor said that if it is bounded and monotonic then it will converge, but what are all the rules or characteristics a sequence must have?

I've been looking at some partial differential equation solving recently, and things like the total derivative come up quite often, lets say, When you can constrain y as a function of x, then [math] \frac{d}{dx} z(x,y) = \frac{dz}{dx} + \frac{dz}{dy} \frac{dy}{dx} [/math] (didn't know how to get the partial operator symbol, but I'm sure you know where it's meant to be) Now I understand that this obviously works, it can be shown via the chain rule, but I have two little intuitive problems with it 1. If you know y as a function of x, why can't you simply substitute in the function of x, and then just have an ordinary differential equation of z(x) 2…

Integration of e^sinx with respect to x. Do you think it ever terminates?

I suspect that everyone does polynomial division the same way, the way we learned in high school. However, this can lead to large exponentially increasing errors. Problems will occur if the denominator has a large magnitude root and the quotient is long. For example, if 1000 is a root of a long polynomial and it is deflated, expect large errors. If the polynomial has high enough degree, expect infinity at some point in the deflated polynomial. For an explanation of why this occurs and what you can do about it see: cnx.org/content/m43398/latest/, "Exponential errors in polynomial evaluation, deflation, & division" P.S. Understanding this article is helped immens…

prove : -0=0

I'm a CS student and the only classes (or class) I had on Calculus was one that only involves getting derivatives and integration with very (very) trivial applications. I seriously doubt that Calculus is JUST that. Anything I should be learning about? Rather can someone point me to more materials?

I'm looking for a numerical stability and error estimation of a finite element approximation of Navier-Stokes equations (with combustion). I define variables and operators on a domain that has both space and time axes (Ω = Ωs x [0,tmax]), so the transport equation looks generally like this div ( u [ v; 1 ] - D [ su; 0 ] ) = Q, where u is either density (of one of the species), velocity, temperature or pressure; v is velocity; D is diffusion coefficient; s is gradient over the space domain Ωs; and Q is either reaction rate, pressure gradient plus buoyancy force (-sp + fb), energy release or 0. The approximation scheme is ui = Σj<i ( D ( dij-1 - di.-1) - …

I need to derive Euler-Lagrange equations and natural boundary conditions for a given model. I've worked out and broken down the model into the following 5 parts: J1 = ∫ {ϕ>0} |f(x) − u+(x)|^2dx J2 = ∫ {ϕ<0} |f(x) − u-(x)|^2dx J3 = ∫ Ω |∇H(ϕ(x))|dx J4 = ∫ {ϕ>0} |∇u+(x)|^2dx J5 = ∫ {ϕ<0} |∇u-(x)|^2dx. where f : Ω → R and u+- ∈ H^1(Ω) (functions such that ∫ Ω(|u|^2 + |∇u|^2)dx < ∞). I need to differentiate each of these 5 equations in terms of ϕ,u+ and u-, any assistance would be very appreciated as I'm weak in calculus. I tried getting the first variation, for example for J1(ϕ), let v be a perturbation defined in a …

I stumbled upon the equation lim n -> infinity (1+(1/n))^n= e. My textbook says that the larger n is, the closer it gets to e. In calculating interest rates, n is replaced by amount of times compounded in one year. Anyone know why the irrational number e is so special? In that (1 + (1 divided by any large number)) an all to the power of the same very large number brings you close to e? This is utterly nonsensical to me. When this is applied to continuous compounding, where the compound amount A for a deposit of P dollars at an interest rate r per year compounded continuously for t years is given by A = Pe^rt. Does anyone know how did they derive the formula A = Pe^…

Hi, [math]3x/(x^2+4)[/math] I want to find the derivative of this but I can't use the quotient rule. So I figure I'll use the chain/product rule. I'm new to calc though. Do I use both? Or just one? [math]= 3x(x^2 +4)^-1[/math] [math]=-3x(2x(x^2+4)^-2[/math] That's the chain rule I think. Do I go on to use the product rule?

I recently was marked down on my homework when solving a problem relating to linear density. I understand the process of computing the mass, but in my mind it doesn't make sense. I would appreciate if someone could explain it to me. The problem similar to as follows: Calculate the mass of a 3 meter rod whose linear density is calculated using the formula p(x) = sin(3x)+5 kilograms per meter. I understand you setup an integral w.r.t. x but in my mind the problem is missing a component. Wouldn't the mass of the rod vary depending on the radius of the rod? I understand that the density is linear, but if the rod was 30 meters in radius, wouldn't it be greater than …