Functions Definition and 1000 Threads

I'm given the following Piecewise function when $f:[0,1]\to[0,1]$: $f(x) = x$ when $x\in\Bbb{Q}$ $f(x) = 1-x$ when $x\notin\Bbb{Q}$ I need to prove that $f$ is continuous only at the point $x=\frac{1}{2}$. For this problem, I know I need to use the fact that a function $f$ is continuous at a...

Homework Statement So the test is to take the determinant (D) of the Hessian matrix of your multivar function. Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point. For D<0 it's a saddle point, and D=0 gives no information. My question is, what happens if fxx=0? Is that...

Homework Statement Prove that sinx+cosx is not one-one in [0,π/2] Homework Equations None The Attempt at a Solution Let f(α)=f(β) Then sinα+cosα=sinβ+cosβ => √2sin(α+π/4)=√2sin(β+π/4) => α=β so it has to be one-one [/B]

Hello, I've been reading about injectivity from Z to N and surjectivity from N to Z and was wondering whether there was some kind of algorithm that could generate these specific types of functions?

Homework Statement Homework EquationsThe Attempt at a Solution I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?

On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...

[FONT=Times New Roman]I would like to know some general properties of the modulo (remainder) function that I can use to rewrite expressions. For example, say we wanted to prove the following by rewriting the right-hand-side: $$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} = \frac{n - n \pmod d}{d}...

Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows: Question: Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula: V = 999.87 −...

I have seen written out in various places (including this forum) the effective potential function that comes from the solutions to the Schwarszschild Geodesic. But I haven't been able to find the effective potential functions for other solutions to Einstein's field equations. Are there...

I was thinking about the connection between fields and particles. For instance the scalar field Φ(x) and the field Φ(x)+a both represent the same scalar particle. Because the action ∫∂Φ∂Φdx^4 is unaltered and the propagator <0|[Φ(x)+a,Φ(y)+a]|0> is presumably the same. What about if we replace...

Hey! :o Find an order $f_1, f_2, \dots f_{30}$ of the functions that satisfies the relations $f_1=\Omega(f_2), f_2=\Omega(f_3), \dots, f_{29}=\Omega(f_{30})$$$\frac{n}{\lg n} , \ \ n^{\lg n} ,\ \ (\sqrt{2})^{\lg n}, \ \ n^2, \ \ n!, \ \ (\lg n)! ,\ \ \left( \frac{3}{2} \right)^n ,\ \ n^3 ,\ \...

Homework Statement For positive integers m, k, and n , let mkn be defined as mkn = kmn , where k\frac {m}{n} is a mixed fraction. What is the value of 643 + 364 ? Homework Equations I attempt the other few similar questions where the solution are as follow 832 + 382 = \frac {169}{24} 641 +...

Hello! (Smile)I want to determine if $\sqrt{n}$ is $\Theta $ / $O$ / $\Omega$, $o$, $\omega$ of $n^{\sin n}$. To do so we could calculate the limit: $$\lim_{n \to +\infty} \frac{\sqrt{n}}{n^{\sin n}}$$ right? But how can we find the limit, although $\lim_{n \to +\infty} \sin n$ does not...

Suppose two people, X and Y, have two different stopwatches. X starts his/her stopwatch as some particle passes an origin. We can model the velocity of the particle by ##\vec{v}(T)##, where ##T## is the reading on the first stopwatch. After an amount of time ##\Delta t##, Y starts his/her...

Homework Statement Show that the equations xy^2+zu+v^2=3 x^3z+2y-uv=2 xu+yu-xyz=1 can be solved for (x,y,z) as functions of (u,v) near the point (x,y,z,u,v)=(1,1,1,1,1) and find dy/du at (u,v)=(1,1) Homework Equations Multivariable calculus differentiation 3. The Attempt at a Solution I...

What are the rules if you have a sequence f_n of real-valued functions on \mathbb R and consider the sequence f_n(x_n), where x_n is some sequence of real numbers that converges: x_n \to x. All I have found is an exercise in Baby Rudin that says that if f_n \to f uniformly on E, then f_n(x_n)...

Homework Statement f(x) = sin5x ; [π/5,2π/5] finding the point c which f'(x) =0. I understand the theorem and how to complete it, my issue is using the triq functions Homework Equations f'(x) = 5cos5x The Attempt at a Solution 5cos5x=0 cos5x=0 5x=π/3 x=π/15 my answer is not correct, I am...

Here are conditions I use to define my problem: 1) I use cumulative distribution of 2 logistic functions g1(x) and g2(x) with g2 is translated to the right of the g1(x) on x-axis. 2) I make a transformation to eliminate both tails of the function which will not have a significant contribution to...

Find all functions $F(x):\Bbb{R}\longrightarrow \Bbb{R}$ such that $F(x)-F(y)\leq (x-y)^2$ for all $x,y\in \Bbb{R}$ Edited for correct a typo.

Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...

For example, say we have ##\frac{x^4(x - 1)}{x^2}##. The function is undefined at 0, but if we cancel the x's, we get a new function that is defined at 0. So, in this case, we have ##x^2(x - 1)##, then ##x^2(x - 1)(1)##, and since ##\frac{x^2}{x^2} = 1##, we then have ##\frac{x^4(x - 1)}{x^2}##...

Hello, I am trying to solve this. This material is not covered in my class, but I still want to know how to do it. If cos(t)=$\frac{-9}{10}$ where $\pi$ <t<$\frac{3\pi}{2}$ find the values of cos(2t)= sin(2t)= cos($\frac{t}{2}$)= sin($\frac{t}{2}$)= Give exact answers, do not use decimal...

Homework Statement [/B] By using the Ascoli-Arzela theorem, show that the functions fn(z) = zn in Δ(1)n = 1, 2,..., are not equicontinuous. Homework Equations [/B] A family F of complex-valued functions on A is called equicontinuous if ∀ε > 0, ∃δ > 0 such that |f(z) - f(w)| < ε, ∀z, w ∈ A...

Homework Statement 1. Find a formula for (f g)(x) = ? 2. Find a formula for (f f )(x) = ? 3. Find a formula for the composition below. g(h(x)) = 4. Find a formula for the composition below. (h g)(x) =The Attempt at a Solution 1. f(g(x)) 2. f(f(x)) 3. (g º h)(x) 4. h(g(x)) Why are these...

I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ... Can someone please suggest (i) an online...

Homework Statement [23/4, 2] 4/(x√(x4-4)) Homework Equations ∫ du/(u√(u2 - a2)) = 1/a(sec-1(u/a) + c The Attempt at a Solution I first multiplied the whole thing by x/x. This made the problem: 4x/(x2√(x4 - 4)) Then I did a u substitution making u = x2. Therefore, du = 2xdx. I multiplied by...

Suppose you have a λφ4 theory. Books only seem to calculate counter-terms for 2-pt and 4-pt functions. But what about 3 particles scattering into 3 particles? Do the counter-terms determined by renormalizing the 2-pt and 4-pt functions cancel divergences in 3x3 scattering? For example, take...

I'm trying to solve this problem from a high school math competition: Find all functions f : R → R such that, f(f(x+y)-f(x-y))=xy, for all real x,y. Any ideas of how to approach it. I have found that f(0)=0, if x=y f(f(2x))=x^2

As I read in the James Stewart's Calculus 7th edition, he said: My question is: Is f(x)\rightarrow 0 the same as f(x) = L? For example, f(x) = x^2 \displaystyle\lim_{x\rightarrow 5}f(x) = 25 I can say that f(x) = x^2 approaches 25 as x approaches 5. Therefore, can I say that the...

I have just posted an edit to my (very) recent post: http://mathhelpboards.com/analysis-50/apostol-continuity-amp-differentiabilty-14190.htmlin the Analysis Forum. I am having trouble with the following Latex expression:\text{lim}_{x \rightarrow c} f^* (x) = \text{lim}_{x \rightarrow c}...

Homework Statement Verify by brute force that the three functions cos(θ), sin(θ)eiφ and sin(θ)e−iφ are all eigenfunctions of L2 and Lz. Homework Equations I know that Lz = -iћ(∂/∂φ) I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged...

I'm a bit confused wether or not there is a link between harmonic functions (solutions of the Laplace pde) and harmonic oscillating systems? What is the meaning of "harmonic" in these cases? Thanks!

Hello! (Wave) The following two functions are given and I want to find their time complexity. function BinarySearchTreeLookUp(key K,pointer R): Type if (R==NULL) return nill; else if (K==R->key) return R->data; else if (K<R->key) return(BinarySearchTreeLookUp(K,R->LC))...

Homework Statement inputs x1(t) = cos(ω1t), x2(t) = cos(ω2t). Show that output g(t) (sum of x1 + x2) = 0.5cos[(ω2-ω1)t] + 0.5cos[(ω2+ω1)t] Homework Equations included in upload of attempted solution. Trig identities. The Attempt at a Solution Uploaded in pdf. A lot more has been done on the...

Hi! (Wave) Find the cardinal number of $C(\mathbb{R}, \mathbb{R})$ of the continuous real functions of a real variable and show that $C(\mathbb{R}, \mathbb{R})$ is not equinumerous with the set $\mathbb{R}^{\mathbb{R}}$ of all the real functions of a real variable. That's what I have tried: We...

Homework Statement Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B] Homework Equations 1. z=a+bi 2. re^itheta The Attempt at a Solution I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...

Homework Statement In 2000 the population of a country was estimated to be 8.23 million. In 2010 the population was 9.77 million. Assume that the number of people P(t) in millions at time t (in years since 2000) is modeled by the exponential growth function. P(t) = Aekt Find P(t) giving the...

Homework Statement Hi! Does anyone know how to solve the inverse of these functions? y=(4x^2+2x-2)/(8x^2-4x+6) y=(x+1)/(x^2) I would appreciate your help with these exercises. The Attempt at a Solution For the first one: 8yx^2-4xy+6y=4x^2+2x-2 For the second exercise: yx^2=x+1 yx^2-x=1

I was wondering if it's possible to use the TI89 Titanium's built-in solver with programs. More specifically, for compressible flow problems, I'd like to calculate mach number based on area ratio, specify whether the flow is subsonic or supersonic, then do something with the corresponding...

Homework Statement The function f(x) = xe-3x2 is expressed as a linear combination of the basis functions un(x), which are orthogonal and normalised from minus infinity to infinity. It is expressed by xe-3x2 = ∑anun(x) the un(x)'s are even functions of x for n = 0,2,4 and are odd functions of...

I have a question concerning how how we define the differentiation and integration operators. Firstly, I know that functions are typically defined as an ordered triple triple ##(X, Y, f)## such that ##f⊆X×Y##, where ##x \in X## and ##f(x) \in Y##. This all seems nice and fine, but we also define...

Homework Statement If we shift a curve to the left, what happens to its reflection in the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c) where f is a one-to-one function. Homework EquationsThe Attempt at a Solution Initially I did this...

So I was watching this video on Khan Academy, and it talks about graphing functions that have values in multiple dimensions. It shows how to represent a linear function in 3 dimensions with a set of vectors, L ={p1 + t(p1-p2)|t∈R} where p1 and p2 are vectors that lie on the line you want to...

$$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$ the ans the TI gave me was $\frac{\sqrt{6}}{4}$ the derivative can by found by the product rule. but really expands the problem so not sure how the $\frac{d}{dx}$ played in this.

Homework Statement For f(x) = abs(x^3 - 9x), does f'(0) exist? The Attempt at a Solution [/B] The way I tried to solve this question was to find the right hand and left hand derivative at x = 0. Right hand derivative = (lim h--> 0+) f(h) - f(0) / h = (lim h--> 0+) abs(h^3 - 9h) / h...

I have been looking at my old calculus textbook because to my dismay I seem to have forgotten most of the calculus I learned. I am given 3 cases of ##(f+g)(x) ##. Case 1 both f and g are even: I know ##f(x) = f(-x) ## and ##g(x)=g(-x) ## for the domain of the function. I can reason by...

Homework Statement Let F be the set of one-to-one functions from the set ##{1,2,..,n}## to the set ##{1,2,...,m}## where ##m \geq n \geq 1##. Then how many functions f in F satisfy the property ##f(i)<f(j)## for some ##1 \leq i \leq j \leq n## Homework EquationsThe Attempt at a Solution...

Hello, I am just doing my homework and I believe that there is a fault in the problem set. Consider the set of functions defined by V= f : R → R such that f(x) = a + bx for some a, b ∈ R It is given that V is a vector space under the standard operations of pointwise addition and scalar...

Hi everyone, I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation. I have tried searching on the web but have had difficulty in finding something which...

Is it okay to define a local operator as an operator whose matrix elements in position space is a finite sum of delta functions and derivatives of delta functions with constant coefficients? Suppose your operator is M, and the matrix element between two position states is <x|M|y>=M(x,y). It...