Functions Definition and 1000 Threads

1. Determine the equation that represents the relationship between the power and the current when the electric potential difference is 24v and the resistance is 1.5 Ω. 2. Draw a graph of the parabola that corresponds to the equation found in (a). 3. Determine the current needed in order for...

Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...

Hello! I'd like to ask for a help about how to compute accurately functions which has very intense oscillations. My example is to estimate $$I = \int_0^{\infty} \sin(x^2) dx= \int_0^{\infty}\frac{\sin(t)}{2\sqrt{t}} dt$$.I tried trapezoid rule over one oscillation at a time, but result is...

Hi PF! I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact...

Hi PF! I have a function that looks like this $$f(r,\theta) = \sinh (\omega \log (r))\cos(\omega(\theta - \beta))$$ You'll notice ##f## is harmonic and satisfies the BC's ##f_\theta(\theta = \pm \beta) = 0##. Essentially ##f## has no flux into the wall defined at ##\theta = \pm \beta##. So we...

When dealing with any tensor quantity, when making a coordinate transformation, we should put a bar (or whatever symbol) over the functions or over the indices? For exemple, should the metric coefficients ##g_{\mu \nu}## be written in another coord sys as ##\bar g_{\mu \nu}## or as ##g_{\bar \mu...

How many coordinate functions of a many-to-1 function must also be many-to-1 ? Let ##F## be a function from ##\mathbb{R}_n## into ##\mathbb{R}_n##. Represented as an ##n##-tuple in a particular (not necessarily Cartesian) coordinate system ##h##, ##F## is given by ##n## coordinate functions...

In which order did the following functions* of organisms evolve, from the very first life through to all extant organisms. Respiratory; digestion/excretion; reproduction; peripheral nervous; endocrine; integumentary/exocrine; circulatory; renal/urinary; lymphatic/immune; skeletal; Muscular...

I have a relatively light question about special functions. As an example, it can be shown that ##\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\sin x} ~ dx = \frac{\sqrt{\pi} ~\Gamma (\frac{3}{4})}{2 \Gamma (\frac{5}{4})}##. Generally, the expression on the right would be taken as "the answer" to...

$\textsf{ 1.3.14 Verify the following given functions is a solution of the differential equation}\\ \\$ $\displaystyle y'-2ty=1\\$ $\displaystyle y=e^{{t}^2}\int_{0}^{t} e^{- s^2}\,ds+e^{t^2}$ \begin{align*}\displaystyle &= \end{align*} ok this one kinda baffled by the $\int$ presume to do...

If we have a relation, ##R##, and it's inverse, ##R^{-1}## they behave such that a point on ##R##, say (a,b), corresponds to the point (b,a) on ##R^{-1}## This is a reflections across the line y=x. This relation does not mean that ##R^{-1}## is a function. For example, Let ##R## be...

$\textsf{ Verify the following given functions is a solution of the differential equation}\\ \\$ $y''''+4y'''+3y=t\\$ $y_1(t)=t/3$ \begin{align*} (t/3)''''+4(t/3)'''+(t/3)&=t\\ 0+0+t&=t \end{align*} $y_2(t)=e^{-t}+t/3$ \begin{align*} (e^{-t}+t/3)''''+4(e^{-t}+t/3)'''+3(e^{-t}+t/3)&=t\\...

Hi there. It is obvious that if you have two differentiable functions ##f(x)## and ##g(x)##, then the product ##h(x)=f(x)g(x)## is also smooth, from the chain rule. But if now these functions are multivariate, and I have that ##h(x,y)=f(x)g(y)##, that is ##f(x,y)=f(x)## for all y, and similarly...

My question is maybe elementary but I don't know the answer. I have a function f absolutely continuous in (a,c) and in (c,b), f continuous in c. Is f absolutely continuous in (a,b)? I think the answer is negative but I can't find a counterexample. I really apreciatte your help.

Homework Statement Consider ##u\left(x\right)=2\left[\frac{-x}{4}\right]## (a) Find the length of the individual line segments of the function, (b) Find the positive vertical separation between line segments. Homework Equations The output of Greatest Integer Functions are always integers. The...

Hi PF! Given a function ##B## defined as $$B[f(x)]\equiv f''(x) + f(x) + const.$$ Evidently in order for this function to be in the real Hilbert space ##H## we know $$const. = -\frac{1}{x_1-x_0}\int_{x_0}^{x_1} (f''(x) + f(x))\,dx.$$ Can someone please explain why? I can elaborate further if...

Homework Statement State the amplitude, period, phase shift, and vertical shift: ##y=\frac{5}{2}sec\left(\frac{π}{x}-4π\right)-2## Homework EquationsThe Attempt at a Solution [/B] Amplitude: Amplitude is equal to the absolute value of a. So the amplitude here is ##\frac{5}{2}## Vertical...

Homework Statement Evaluate and express your answer in radians: $$cot^{-1}\left(1\right)$$ Homework EquationsThe Attempt at a Solution I start by identifying that the domain of Arccotangent is all real numbers. So 1 is in the domain. From here, I looked at the unit circle and saw that...

How to prove that essentially bounded functions are uniform limit of simple functions. Here measure is sigma finite and positive.

The circle ## x^n+y^n=1 ##, for n integer >2 in a metric space with distance function: ## \sqrt[n] {dx^n+dy^n} ## has corresponding trigonometric Sine and Cosine functions defined in the usual way. Finding the sine or cosine of the sum of two angles, derivatives and curvature of a line in such...

I'm getting a bit lost on some of the basics. So a Likelihood function determines the plausibility of parameters given the observed random variables. This is fine and all, but something seems a bit off. The observed random variables themselves must be generated from a probability distribution...

I have a question regarding different functions. Suppose we have two functions ##f## and ##f'## with same domain, but different codomains. Consider that ##f': x' \mapsto f'(x')## and ##f: x \mapsto f(x)##. If ##x' = x + \sigma##, with ##|\sigma| << 1##, can we say that ##f'(x') \approx f(x')##?

Hi PF! One way to solve a simple eigenvalue problem like $$y''(x)+\lambda y(x) = 0,\\ y(0)=y(1)=0$$ (I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...

Helo, given ##f:R^n\rightarrow R^m## and ##g:R^m\rightarrow R^e## both class ##C^m##. Is the composition ##g\circ f## of class ##C^m## ?.

I started studying distribution theory and I am struggling with the understanding of some basic concepts. I would hugely appreciate any help, made as simple as possible, because by now I'm only familiar with the formalism, but not all the meaning behind. The concepts I am struggling with are...

So if i take the rules that a straight vertical line drawn through the function with more than one intersection implies it is not a function, to mean that the quadratic equation for a circle is not a function. Furthermore, it also implies a cubic equation, such as x^3 can be a function, because...

Today, while reading about bijections, a question came into my mind. And that is: is there any way that two different functions ##f## and ##g## acting on a same point ##p## gives the same output? In symbols, as I'm not good in English, is it possible that ##f (p) = g(p)## with ##f \neq g##?

Homework Statement a) I have to find and expression for sequence of $b_n$ in terms of generating functions of the sequence of $a_n$ $$b_n = (-1)^{n}(n+1)a_0 +(-1)^{n-1}n a_1+...+(-1)2a_{n-1}+a_n$$ with $$a_n = a_{n-1} +8a_{n-2} -12a_{n-3} +25(-3)^{n-2} + 32n^2 -64$$ b) I have to use the...

Homework Statement ∫ e1000((sinx)/x) dx [0 to 1000 : bound of integration]. Solve this integral of a sharply peaked function without a calculator. Homework Equations I'm doing this in relation to statistical thermodynamics - I think I need to use Sterling's Approximation or a gamma function...

So, in Physics, we were to learning about linearising data of a practical. We know that if the graph of the data of a practical or experiments represents a surd function (y=x^0.5), then it can be linearised by graphing y^2 against x. Therefore, y^2 would be directly proportional to x. However...

The literature mentions "functions that are effectively computable in the informal sense". What is meant by that? It would be helpful to have an example involving "informal sense" vs. "formal sense" for some numerical function. All help appreciated. am

Hello all, I am trying to solve the following problem: In the given graph, we see the function: \[f(x)=ka^{-x} , x\geq 0\] 1) Find k and a 2) Find x1 3) Show that an increase of 2 units in x brings a 50% reduction in the value of the function f. I have tried solving it, but taking two...

My textbook says that if ##X: \Omega \to \mathbb{R}## is discrete stochast (I.e., there are only countably many values that get reached), then it suffices to know the probability function ##p(x) = \mathbb{P}\{X =x\}## in order to know the distribution function ##\mathbb{P}_X: \mathcal{R} \to...

Homework Statement Homework Equations As the book says , an affine function of a line is A\rightarrow \mathbb{R} and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is...

I know that some functions are ## 2 \pi ## periodic but what does it mean that a function is ##1##-periodic. Is it ##f(x+1n) = f(x)## where ## n \in \mathbb{Z} ## ?

Given a function in ##f \in L_2(\mathbb{R})-\{0\}## which is non-negative almost everywhere. Then ##w-lim_{n \to \infty} f_n = 0## with ##f_n(x):=f(x-n)##. Why? ##f\in L_2(\mathbb{R})## means ##f## is Lebesgue square integrable, i.e. ##\int_\mathbb{R} |f(x)|^2 \,dx< \infty ##. Weak convergence...

Hi PF! Can someone explain the second line of the proposed solution on this thread to me https://mathematica.stackexchange.com/questions/138919/how-to-implement-a-loop-inside-piecewise Specifically, I have a function un(x) that looks like I am trying to make this function piecewise such that...

Homework Statement Find the derivative of the following functions. h(x)=8x2√x2+1 Homework EquationsThe Attempt at a Solution I just had a lesson on friday but it was a bit confusing to me. I am able to solve some problems but ones that contain multiple rules confuse me. If someone could...

The question wants (f•g)(x). I understand this to be f(g(x)). This means to plug the value of g(x) into every x I see in f(x) and simplify. f(3x^2) = 4(3x^2) + 7 f(3x^2) = 12x^2 + 7 So, f(g(x)) = 12x^2 + 7. This is not the book's answer.

Hello everyone. Time to get back to math. I have forgotten how to find asymptotes of rational functions. I think there are three types of asymptotes. Can someone show me how to find asymptotes of rational functions? What exactly is an asymptote?

I am trying to show that that the Airy functions defined below satisfy: $W[Ai(x),Bi(x)]=1/\pi$. $$Ai(x)=\frac{1}{\pi} \int_0^\infty \cos(t^3/3+xt)dt$$ $$Bi(x)=\frac{1}{\pi}\int_0^\infty \bigg[ \exp(-t^3/3+xt)+\sin(t^3/3+xt)\bigg]dt $$ I tried to compute it directly but I got stuck, here's the...

I am in the trigonometry section of my precalculus textbook by David Cohen. In Section 6.2, David explains how to evaluate trig functions without using a calculator but it is not clear to me. Sample: Is cos 3 positive or negative? How do I determine if cos 3 is positive or negative without...

where rk are the roots of , Find S. -------------- I don't have a decent approach. I am very bad at summations. Although, my friend asked me to assume complex analysis and I still couldn't do it. A simple hint to this would be appreciated. Peace.

Define the numbers $e_k$ by $e_0 = 0$, $e_k = \exp(e_{k-1})$ for $k \geq 1$. Determine the functions, $f_k$, for which \[f_0(x) = x, \;\;\;\;f’_k = \frac{1}{f_{k-1}f_{k-2}\cdot\cdot\cdot f_0}\;\;\;\; for\;\;\; k \geq 1.\] on the interval $[e_k, \infty)$, and all $f_k(e_k) = 0.$

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with the proof of Proposition 2.2.9 ... ... Duistermaat and Kolk's Proposition 2.2.9 read as follows: In the above text...

Differentialbility & Continuity of Multivariable Vector-Valued Functions ... D&K Lemma 2.2.7 ... I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Lemma 2.2.7 (Hadamard...) ... ... Duistermaat and Kolk's Lemma 2.2.7 and its proof read as...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 4 on page 66 ... Kantorovitz's Example 4 on page 66 reads as follows:In the above example, Kantorovitz...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 4 on page 66 ... Kantorovitz's Example 4 on page 66 reads as follows: In the above example, Kantorovitz...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ... Kantorovitz's Example 3 on pages 65-66 reads as follows...