Function Definition and 1000 Threads

Homework Statement: Trying to understand electrical calculations in AC Relevant Equations: I = V/R Here it is :

Hi there, I was wondering how computer programs such as geogebra or more advanced packages such as mathlab or wolfram plot graphs for functions. Do they calculate values and interpolate? Do they take derivates to determine maxima and turning points? Do they have any other way to somehow...

İf $$f:\mathbb{R^n}\to \mathbb{R}$$ then $$\nabla f:\mathbb{R^n}\to \mathbb{R^n}$$ $$x\to \nabla f(x)$$ is true?

On the book, it says, "Let ## f ## be defined by ## f(4n)=f(4n+1)=0, f(4n+2)=2 ## and ## f(4n+3)=1 ##, for all integers ## n ##". (Other answers are possible). But I don't understand, how does this work in the problem? I know that it must has something to do with the period, which is ## 4 ## in...

I am looking for some academical concept to work on 3 parts : 1) Real and imaginary analysis of two functions describing 2 events 2) If the first event's function is imaginary and the second is real , how can we analyse the intersection that show how the imaginary function turned out...

So I know that since ##x \in R## that means ##2x## can achieve all possible values on the real number line meaning ##f(x)## is a constant function. And I know hwo to calculate the limit beyond that. However my teacher made a point which I dont necessarily agree with he said, if ##f(x)## wasn't...

TL;DR Summary: Continuity of a function, Calculus newbie, delta, epsilon, Greetings! I have just started studying Calculus for my engineering course, and I am already facing some problems to understand the fundamental ideas regarding the continuity of a function. I'd be very much grateful if...

Hi, I try and solve this problem I have solved the problem in different parts But me not sure how to plot the graph. Maybe someone knows? Merci

I simplified this function to ##\frac{1}{2} (\frac{x tan^3(x)} {(sin²x)²(1-tan²x)}## Now further can I not write ##1-tan²x## as ##\frac{cos2x} {sin²x}## ? If I do that I get ##\frac{1}{2} (\frac{x tan^3(x)} {sin²x cos2x}## On graphing this on desmos I get two different graphs for these...

Hello! Consider this transfer function $$ G^\#(q) = q \cdot \frac{q^2 + 2q - 3}{q^2 - 25} $$ a) For which values of Ta > 0 is the G(q) step response capable? b) For which values of Ta > 0 is the G(q) realizible? c) Is it possible to find a sampling time Ta > 0 so that the G(q) is BIBO stable...

Hi, I'm not sure if I've solved the problem correctly In order for the Implicit function theorem to be applied, the following two properties must hold ##F(x_0,z_0)=0## and ##\frac{\partial F(x_0,z_0)}{\partial z} \neq 0##. ##(x_0,z_0)=(1,2)## is a zero and ##\frac{\partial...

Hi, I'm not sure if I've understood the task here correctly For the Implicit function theorem, ##F(x,y)=0## must hold for all ##(x,y)## for which ##f(x,y)=f(x_0,y_0)## it follows that ##f(x,y)-f(x_0,y_0)=0## so I can apply the Implicit function theorem for these ##(x,y)##. Then I can write...

For this problem and solution, I'm confused how ##x \in (c - \delta, c + \delta)## is the same as ##0 <| x - c| <\delta##. I think it is the same as ##c - \delta < x < c + \delta## which we break into parts ##c - \delta < x \implies \delta > -(x - c)## and ##x < c + \delta \implies x - c <...

The ####x partial derivative is equal to $$L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ and the partial for ##y## is $$L \frac{4y}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ Using the limit definition of partial derivatives I got the partial wrt ##x## is $$L \frac{h^{\frac{4}{5}}}{h}$$ which doesn’t exist...

Dear everyone, I have a question on how to show that an integral is divigent. Here is the setup: Suppose that we have the following function ##\sigma(x)=\frac{1}{x^{2-\varepsilon}}## for an arbitrary fixed ##\varepsilon>0.## \begin{equation}...

So say I have to find the x intercept of this function $$log_{10}(x²)$$ I get x={-1,1}. But if I try to find the x intercept of this same function after simplifying I get $$2log_{10} (x)$$ I get x={1}

For this problem, I'm confused by the implication from the antecedent ##0 < |x - c| < \delta## to the consequent. Should the consequent not be ##|f''(x) - f''(c)| < \frac{1}{2}## where ##\epsilon = \frac{1}{2}## (Since we are applying the definition of a limit for the first derivative curve)...

For this problem, However, I'm confused how their got their solution. My solution is, using set builder notation, ##[ (x,y) \in \mathbf{R} : 1 - \cos x + y^4 ≥ 0 ]## which implies that ##V(0,0) = 0## so it satisfies the first condition for being sign definite, sign semidefinite, and sign...

For this problem, My solution is ##P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0## where n is a member of the natural numbers Base case (n = 1): ##P(x) = a_0x^0 = a_0## Thus ##\lim_{x \to \infty} \frac{P(x)}{e^x} = \lim_{x \to \infty} \frac{a_0}{e^x} = a_0 \lim_{x \to \infty}...

Let ##f## be a continuous function defined in ##\mathbb{R}^n##. ##||\cdot ||## is the standard Euclidean metric. Then here are my suggested ways to choose ##f##: 1. Choose any continuous ##f## that satisfies $$1=\sup_{||x||\leq 1}||f||\neq \max_{||x||\leq 1}||f||$$ because the inequality...

Hi, I don't know if I have solved task correctly I used the epsilon-delta definition for the proof, so it must hold for ##f,g \in (C^0(I), \| \cdot \|_I)## ##\sup_{x \in [a,b]} |F(x)-G(x)|< \delta \longrightarrow \quad |\int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx |< \epsilon## I then...

For this problem, I am trying to prove that this function is non-differentiable at 0. In order for a function to be non-differentiable at zero, then the derivative must not exist at zero ##⇔ \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}## does not exist or ##⇔ \lim_{x \to 0^-} \frac{f(x) - f(0)}{x...

For this problem, The solution is, However, I'm confused how ##0 < | x - 1|< 1## (Putting a bound on ##| x- 1|##) implies that ##1 < |x+1| < 3##. Does someone please know how? My proof is, ##0 < | x - 1|< 1## ##|2| < | x - 1| + |2| < |2| + 1## ##2 < |x - 1| + |2| < 3## Then take absolute...

For this problem, THe solution is, However, does someone please know why from this step ##-1 \leq \cos(\frac{1}{x}) \leq 1## they don't just do ##-x \leq x\cos(\frac{1}{x}) \leq x## from multiplying both sides by the monomial linear function ##x## ##\lim_{x \to 0} - x = \lim_{x \to 0} x= 0##...

I am trying to solve (a) and (b) of this question. (a) Attempt We know that ##\frac{2}{3} < \frac{e(t) - e(s)}{t - s} < 2## for ##t \neq s \in (c(-d), c(d))## Thus, taking the limits of both sides, then ##\lim_{t \to s} \frac{2}{3} < \lim_{t \to s} \frac{e(t) - e(s)}{t - s} < \lim_{t \to...

How Can I add bar Legend for g[t] in the following code in mathematica? Thanks in advance? f[t_] := t + 1 g[t_] := t^3 + 3*t + 12 ParametricPlot[{f[t]*Cos[\[Theta]], f[t]*Sin[\[Theta]]}, {t, 0, 1}, {\[Theta], 0, 2*Pi}, ColorFunction -> Function[{x, y, t}...

Hi, I have problem to prove that the following inequality holds I thought of the following, since it is a convex function and ##x_1 < x_2 <x_3## applies, I started from the following inequality ##f(x_2) \leq f(x_3)## and transformed it further $$f(x_2) \leq f(x_3)$$ $$f(x_2)-f(x_1) \leq...

So I have: ##H(\omega)=(\exp(-i\omega)-\exp(i\omega))\exp(i\omega)##, I denote by ##Z(\omega)=\exp(i\omega)##, to get: ##H(\omega)=Z(-\omega)Z(\omega)-Z(\omega)^2##, now, I want to find ##h[n]##, I think it should be: ##h[n]=z[-n]*z[n]+z[n]*z[n]##. But I am not sure how to calculate the...

This is the component for Authors to Login to the Web Application import { Button, CircularProgress, Fade, Link, TextField, Typography } from '@material-ui/core'; import { ThemeProvider, createTheme, makeStyles } from '@material-ui/core/styles'; import axios from 'axios'; import React, {...

I first solved for the extension of the spring when its at equilibrium w/ the mass. I got Δx = 0.122m. I thought that if I added this with 0.1m, this would give me my amplitude. I then set 0.5m as my c value and plugged the rest of my values in from there. What am I doing wrong?

Hi, I am doing my thesis on quantum entanglement and I don't seem to wrap my head around what really happens to an entangled system during a local measurement. I happen to know that information can't travel faster than light I could believe that the collapse of the wave function wouldn't allow...

For F: X x I-->Y, defined by F(x,t) = y, next define G: Y x I-->X by G(y,u) = x. Then for t = u, we have F[G(y,t),t] = F{G[F(x,t),t]}, which will ideally be ##\mathbb{1}##. Given Hatcher's definitions pp. 2-3, to me it's not clear how to "invert" a homotopy without an inverse function--let...

On page 3 of the lecture notes for Stochastic Analysis, it says '##B(s,t)## is the covariance function ##\mathbb{E}[X_sX_t]-\mathbb{E}[X_s]\mathbb{E}[X_t]##. Then On page 5, it says the notes also say that 'the covariance function ##B(s,t)## of a strongly stationary stochastic process is...

Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?

Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?

I read on a post here titled 'Understanding Fourier Transform for Wavefunction Representation in K Space' that if one represents the squared-amplitude as a ratio of differentials, the solution is given. Letting the squared-amplitude be ##\phi##. $$\frac{d\phi}{dp}=\frac{d\phi}{dk}\frac{dk}{dp}$$...

The problem and solution are posted... no. 8 I may need insight on common difference ... In my lines i have, Let the roots be ##(b), (b-1)## and ##(b+1)##. Then, ##x^3-3bx^2+3cx-d = a(x-b(x-b+1)(x-b-1)## ##x^3-3bx^2+3cx-d= a(x^3-3bx^2+3b^2x-x-b^3+b)## ##a=1##. Let...

Refreshing... going through the literature i may need your indulgence or direction where required. ...of course i am still studying on the proofs of continuity...the limits and epsilons... in reference to continuity of functions... From my reading, A complex valued function is continous if and...

If we consider ##v=-3t^2## then: $$x=-t^3$$$$a=-6t$$ Using ##t=-x^{1/3}## we have : ##a=-6(-x^{1/3})=6x^{1/3}##. My answer suggust that ##F=Ax^{1/3}## but in options we have ##F=-Ax^{1/3}##. Can someone guide me where my mistake is?

Attempt : Let me copy and paste the problem as it appeared in the text. Please note that the given problem appears in part (b), which I have underlined in red ink in this way ##\color{red}{\rule{50pt}{1pt}}## Clearly the domain is ##\boxed{\mathscr{D}\{f(x)\}...

I could only verify this for a few elementary functions. Does a proof exist? Does it go beyond the realms of high school mathematics? Many thanks.

$$y = f(x) = \sqrt{9-x^2}$$ According to me, Domain: $$ 9-x^2 \geq 0 \implies (x+3)(x-3) \leq 0 \implies x \in [-3,3] $$ which is correct, but this is how I calculate the range: $$y = \sqrt{9-x^2} \implies y^2 = 9-x^2 \implies x^2 = 9-y^2$$ Now, since $$ 9-x^2 \geq 0 $$ we get $$9-9+y^2 \geq 0...

Am refreshing on this, For the domain my approach is as follows, ##(f-3g)x = f(x)-3g(x)## ##=x-3-3\sqrt{x}##. The domain of ##f-3g## is given by ##f∩g = [{x: x ≥0}]## We have ##y= x-3-3\sqrt{x}=(\sqrt x-\frac{3}{2})^2-\dfrac{21}{4}##. The least value is given by...

Hi, I am having a hard time trying to solve this question. How do I find the local inverse at x0? f (x) = x^4 − 4x^2 Find an expression for f^−1 for f at the point x = −2. Thanks a lot! I would really appreciate any help!!

i need to write a function for DPI screen scaling, so the parameters is from 100 (percentage) to 350 (percentage) and increases at 25 (percentage) increase, it will subtract additional 1 DPI so for example: 100% = 96 DPI which is -4 125% =120 DPI which is -5 150% =144 DPI which is -6 175%...

Attempt : The domain of the function ##\sin(3x^2+1)## is clearly ##x\in (-\infty, +\infty)##. The values of ##x## go into all quadrants where the ##\sin## curve is positive and negative. Hence the range of the function ...

I operated by placing ##S## and ##T## to two singlets belonging to ##X## and therefore established that for ##T, S \in X##, therefore ##f (T) = f (S) \implies S = T##, consequentially: $$f (T \cap S ) = f (T \cap T) = f (T) \cap f (T) = f (T) \cap f (S)$$. I would like to know if my procedure is...

In the following I ask WA to solve the given equation and it produces a solution using the Lambert W function. I thought : $$W(x*e^x) = x$$ but here it seems $$W_n \left(\frac{-MT}{P}*e^{\frac{-MT}{P}}\right) \neq \frac{-MT}{P}$$ Is there a difference between ##W(x)## and ##W_n(x)## ?

Is there a function that outputs a 1 when the input is a multiple of a number of your choice and 0 if otherwise. The input is also restricted to natural numbers. The only thing I can come up with is something of the form: f(x) = [sin(ax)+1]/2 but this does not output a 0 when I want it.

Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.