Analysis and Calculus

I read a few articles, and I already know how fractals themselves work, but I don't get how I'd write a formula for them or make them on paper. Basically all I know about them mathematically is that in a Julia set it just repeats a function over and over again, and a Mandelbrot set uses imaginary numbers, and the initial points don't escape to infinity.

Not sure what I'm doing wrong, but I can't seem to get the integral for the third and fourth piece. I might be using the wrong values for [a,b]. Anyone know the answers and what intervals should be used? Am I supposed to add the first two parts into the third?

Basically that, I know that there has to be some way to infinitely add types of sine waves so that they have the same period, but I don't really know how to do it, and websites arn't discrete enough for me.

Are there formulas for different shapes on a plane other than a circle? What do they generate? Or is that how you make different waves?

Hi everyone I'm new to these forms. I have attached a pdf of a writing project I'm working on in my Calc 3 class. I'm just trying to find somewhere to start, because I'm kind of lost. Most of the Calc 3 class has been based on multivariable calculus and objects in 3-d. The problem states that I must determine how much solar energy will be lost because of the shadows. I must calculate the fraction of energy being lost. We have not studied the loss of energy. I've tried searching google and I have only found complex equations. Is there a simple formula for energy lost? One way I was thinking about solving this problem would be to first calculate the time it t…

I am currently reading the book "e: The Story of a Number" by Eli Maor. And I got stuck at something. In chapter 7 of the book, the author described the method Fermat used to calculate areas under curves of the form [imath]y = x^n[/imath], where n is a positive integer. I am quoting the relevant bit here (sorry, I can't show the figure, but from the description, you can easily receate it): Now I can't get to the final formula. The areas of each rectangle I found are [imath]a^{n+1}, (ar)^{n+1}, (ar^{2})^{n+1},[/imath] and so on. Their sum, [math]A_{r} = a^{n+1} + (ar)^{n+1} + (ar^{2})^{n+1} + \cdots[/math] [math]= a^{n+1}\left(1 + r^{n+1} + r^{2(n+1)} …

So if I graph 3-D patterns like a ripple in a pond or sound waves, instead of it involving something like sin(x^3) and sin(z^3) its sin(x^2+y^3). But how could actual movement 3-D movement in 4 dimensional space be built out of the warping of 2 dimensional things? How could a 2-D thing actually move in a 3 dimensional way? Does x^2 not actually have anything to do with 2 dimensions, is it just coincidence that that's how you find a square or is x^3 coincidentally a unit for a cube? What about x^4units^4? Shouldn't that be a 4th dimensional object? Is space somehow a 2-D plane in every direction? And how is that different than just a plain 3-D object?

Just what the question says. What the heck about electrons and other sub-atomic particles makes them exactly correspond to a dot moving around a circle of infinite right triangles? Or sound waves even?

Does anyone whether parabolic PDEs of the form:<br> [math] u_t(x,t) = -a \cdot x(1-x)^2 u_x(x,t) + \frac {1} {2} b \cdot x^2(1-x)^2 u_{xx}(x,t) [/math] <br>(a, b constants, t > 0, 0 < x < 1) have a closed-form fundamental solution?

Hi I need help to determine if the text books answer is wrong. 6th Edition of Calculus Early Transcendentals by James Stewart if you have the book. Its on page 636 question 3. It states: Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x= t^2 + t , y= t^2 - t ; t = 0 It is basic but the answer they give does not make any sense to me. They found the equation of the tangent to be (2sint*cost)/(lnt+1) I found the equation of the tangent to be y=-x I've drawn the curve and it seems to match my answer at t = 0 If somebody could please confirm my answer or tell me if I'm making some kind of er…

Imagine a world were literally nothing existed, something never came into existence and there is just infinite nothing, well you then realized based on maths that even nothing must produce something because: 1 divided by 0 = Infinity thus 0 x Infinity = 1 and 0 = nothing & 1 = something thus 0 x Infinity = Infinite nothing which equals to 1 thus eternal nothing = 1 something just as Infinity x 0 = 1 so in other words eternal nothing must produce 1/Infinite the size of the nothing of something, thus nothing must create something. Although its hypothetical, if nothing existed in fact, then maybe this equation could be the reason for everyth…

Calculators can do it, why not us?

x/0=undefined; x>0 Undefined= indefinite form or value. Because zero is zero or the absence of value (and all this other number theory crap), it can go into any non zero number as many times as it can, but because of the fact that it "can" go into a non-zero number any amount of times, however many times it actually goes into something is undefined or cannot be kept track of. Opinion: If something doesn't define how many times 0 goes into a non-zero number, it simply does it infinite or at least indefinite times, which explains why mass in 0 volume = infinite density, because math wise it would be x/0; x>0. And then, what if you do in fact define it? Say…

I heard it's for some reason mathematically impossible to have a formula for a polynomial beyond a quartic for some reason, but its a consistant pattern as the degree goes up so I don't see why it wouldn't be possible. Basically I'm looking for a formula that generalizes polynomials and I hear that's impossible for some reason.

I guess its really just a by product of how our math works, but If I swing on the swing and I graphed the position off the ground, not over time, just distance from the ground, it would be a parabola with an exact equation, but no matter what, I would never actually be at 0 distance from the ground yet I could still get imaginary numbers for 0s. And then another problem with it is you can generate real numbers from imaginary numbers by raising them to an exponent, or perhaps there actually is something multiplied by itself to give a quantity that is less than 0 that we don't know of.

............ got my answer from someone else.

The following is taken from "An Introduction to the Theory of Numbers" - G. H. Hardy & E. M. Wright; 6th Edition, page 62 Theorem 57. If [math] (k,m) = d, [/math] then the congruence (5.4.1) [math] kx \equiv l ( mod \;m ) [/math] is soluble if and only if [math] d|l [/math]. It has then just [math] d [/math] solutions. In particular, if [math] (k,m)=1 [/math], the congruence has always just one solution. I'm having a hard time with the proof that "It has then just [math] d [/math] solutions." The following is clear up until the last point: If [math] d|l [/math], then [math] m = dm', \; k=dk', \; l=dl' [/math], and the congruence is equivalent to (5.4…

I followed a link recently by ajb http://en.wikipedia....wiki/Propagator . There was notation in the work in the position space part. In the solutions, you can see there is an imaginary part in the denominator; [math]\pm i\epsilon[/math]. Now my memory is vague, but there was a reason why equations like this would have an imaginary part in the denominator.... of course, someone figured this out.... it involved a problem with a certain equation which could be fixed by assuming part of the denominator was imaginary... but who it was who assumed this and what equations it referred to is escaping me now. I wanted to look more into this, but I can't get anywhere further to…

... doesn't matter.

Prove that (0,2) is open in [math](R^2,d)[/math] where d is the discrete metric: .....................................................d(x,y) = 0 ,if x=y........................................................... ......................................................d(x,y) =1 ,if [math]x\neq y[/math]..................................... Obviously we have to prove that: for all xe(0,2) ,there exists ε>0 such that B(x,ε) is a subset of (0,2) ,or for all xε(0,2) ,there exists ε>ο such that yεB(x,y) iplies [math]0<y_{1}<2[/math],where x =[math](x_{1},0)[/math] and y=[math](y_{1},y_{2})[/math] AM I correct??

So you have a right cone with a certain opening angle, shouldn't there be an equation to find the height for a certain diameter? Seeing as the sides move away at a constant rate there should be some sort of equations realting angle to diameter.

Ok, so I'm doing synthetic division for a polynomial and I have to show all my work, but no matter what number I use from the ration root theorem, nothing works, which means all the 0s are imaginary, but then how do I find out exactly which imaginary numbers there are? Because I couldn't find a "quartic" formula and I'm not always dealing with a polynomial where there's an automatic formula for it like a quadratic formula, so is there still some quick way I can do this? Preferably using synthetic division or should I just say in an equation like that where the highest degree is in x^n, there's n "i"s?

I am not able to solve some questions so I decided to post them here. I got answers to some of them and I just want you to check them. But some others I don't understand at all. NOTE : PLEASE DO THE QUESTIONS ONLY IF YOU ARE 100% SURE Here are the questions. #1 - #5 ** ANSWER PROVIDED CHECK IT PLEASE ** 1) Suppose that x^2-6x-3 <= f(x) <= sqrt(9-x^2) on [0,3]. Give upper and lower bounds for lim x->0+ f(x) and lim x->3^- f(x) lim (x^2 - 6x - 3) = -3 as x --> 0+ lim sqrt(9 - x^2) = 3 as x -->0+ -3 = f(x) = 3 lim (x^2 - 6x - 3) = -12 as x --> 3- lim sqrt(9 - x^2) --> 0 as x --> 3- -12 = f(x) = 0 (What is t…

Newton devised some way to measure the area of a blob by using some infinite triangle theorem, but I have no idea what the theorem is right now.

I was wondering whether someone would be able to solve this equation by getting the maxima and minima and show the working: -3cos(4x-pi) I would attempt it, but i'm not quite sure how to go about it...thanks.