Root Definition and 918 Threads

Hi. I have been trying to calculate the real definite integral with limits 2π and 0 of ## 1/(k+sin2θ) ## To avoid the denominator becoming zero I know this means |k|> 1 Making the substitution ##z= e^{iθ}## eventually ends up giving me a quadratic equation in ##z^2## with 2 pairs of roots...

Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...

Hi I tried like this. ##σ^2=<(λ_1+λ_2+,,,+λ_i)^2>=<λ_1^2>+<λ_2^2>+,,,<λ_i^2>## And I know ##σ^2=Σ_in_iλ_i^2##from equation (4-12) (so this is cheat 😅). So I know also ##<λ_i^2>=n_iλ_i^2##, But why?? I know if I take ##λ=1 ,σ^2=n##,But I don't understand ##λ≠1## version. Sorry my bad...

What is the notation to show element-wise square root of a vector or matrix?

Summary: I want to basically multipartition my pc and use MY android OS copied from my phone as the dual boot OS. Hi all, it's been awhile since I've posted. Sorry for AWOL. What I want to do: Partition my hard drive and run my exact copy of android on the other partition. Why: I want to...

From my understanding, the root locus is only concerned with open loop gain. I figured this means you would ignore the negative feedback loop and calculate the root locus from just the plant's function Workings: zeros: -1 poles: 0, -2, -2, relative degree = 2 => 90-degree asymptotes meeting...

I don't understand this. a is not suppose to be -1; this is the only rule in the equation The answer is the second picture, I just don't know the steps that lead to that answer.

I am going through this article, and it mentions a Kronecker made a discovery about odd prime degree polynomials that makes Abel's Theorem on the impossibility of the quintic easier to prove: https://hubpages.com/education/Abels-Proof--A-Gentle-Introduction-to-the-Sublime-Beauty-of-Mathematics...

Hi! $$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$ $$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$ I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty...

Because it holds that ##\displaystyle\int_{1}^{x}\frac{dt}{t} = \log x##, and ##\displaystyle\int_{1}^{x}\frac{dt}{t^a} = \frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)\hspace{20pt}##when ##a>1## it could be expected that ##\displaystyle\frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)...

I have the following equation: x^2 - 2(m+1)x + 3m + 1=0 Also, I know that x1 is the lowest root of this equation. I need to solve lim (x1) as m->infinity A. 1 B. 3/2 C. 0 D. -1/2 E. -1 I tried to solve the equation with the discriminant then to calculate the limit but didn't work. Also, I think...

Homework Statement Hi everyone, I'm currently making my way through Spivak's calculus and got stuck in question 41 of chapter 5. It's important to note that at this point, the book has only reached the subject of limits (haven't reached continuous functions, derivatives, integrals, series...

Hi guys! Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay? In general, when we are solving power...

Homework Statement I read the expression E=fRT/2 where E is internal energy of ideal gas and f is degrees of freedom, and ##V_{rms} = \sqrt{\frac{3RT}{M}}## Since internal energy for an ideal gas is purely kinetic (according to KTG) I can write 1/2 mv^2 = fRT/2. Now H2 is a diatomic molecule...

Homework Statement Simplify √(53 - 8√15) Homework Equations Numbers can be represented as √a - √b The Attempt at a Solution I had tried to make in an equation where the 2 expressions were equal, but after squaring both sides, I didn’t really know what to do. I had also tried to use something...

Homework Statement "Show that for some ##x\in ℝ##, that ##x^5+2x^4+3x^3+2x^2+x=1##." Homework EquationsThe Attempt at a Solution Okay, so I know from Descartes' rule of sign that the function ##f(x)=x^5+2x^4+3x^3+2x^2+x-1## has exactly one positive root, since the sign of the coefficients...

Is the square root of 945 irrational? I feel it is rational because my TI-84 Plus converts it into 275561/8964, however, I am unsure whether the calculator is estimating. Can someone please advise. It can be broken down into 3√105, and again, my calculator is able to convert √105 into a...

Homework Statement Homework Equations Rolle's Theorem: If f(a)=f(b)=0 then there is at least one a<c<b such that f'(c)=0 The Attempt at a Solution $$y=2x^3-3x^2-12x-6~\rightarrow~y'=6x^2-6x-12$$ The function: y': How do i know y' isn't 0 somewhere? if it's continuously descending, so i...

Problem : A ball is let down a ramp on top of a table with initial velocity of 0 ms-1. When it reaches the end of the ramp, it is launched horizontally. Knowing that we don’t take air resistance or friction into account, and that the height of the ramp is h1, and that of the table is h2...

A ball is let down a ramp on top of a table with initial velocity of 0 ms-1. When it reaches the end of the ramp, it is launched horizontally. Knowing that we don’t take air resistance or friction into account, and that the height of the ramp is h1, and that of the table is h2 relative to ground...

Homework Statement (x.y)ER+ that means x and y >=0 Homework Equations Prove that n√(x+y)<=n√x + n√y The Attempt at a Solution

I have been debating this issue for days: I can't find a recursive function of this equation: ##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}## Starting value 2 always added with pi has been trying to find a solution this for days now, is what I have achieved so far: This...

Homework Statement Solve the ##\lim_{n \rightarrow +\infty} \sqrt [n] \frac {n²+1} {n⁷-2} ## 3. The attempt of a solution: First I thought about using L'Hopital's rule, but the nth root makes it useless. Then I thought about to eliminate the root multiplying it by something that is one, but...

I've recently started my new RA position, and I've been given the task of analyzing a root data file. I'm not completely lost, but I don't exactly know what I'm doing. The point of my post is not to ask for answers, merely advice. A place I could go for info on data analytics. Pointers on how to...

Why is it that for a 7th degree polynomial, the number of real roots is either 1, 3, 5, or 7?

In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this: \frac{d^2...

I'm trying to find ##\sin (\arccos x)##. I let ##\theta = \arccos x## and then use ##\sin ^2 \theta + \cos ^2 \theta = 1##, I get ##\sin (\arccos x) = \pm \sqrt{1 - x^2}##. I'm not sure whether to take the positive or negative root. On Wolfram Alpha is shows that the result is the positive root...

Homework Statement ##\displaystyle \int \sin x \sqrt{1+ \tan ^2 x} dx## Homework EquationsThe Attempt at a Solution So clearly we have that ##\displaystyle \int \sin x \sqrt{\sec ^2 x} dx##, but I am not sure how to proceed. Isn't it true that ##\sqrt{\sec ^2 x} = | \sec x|##? How would I...

I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their...

Hi, Sorry for this weird question. My computer crashed and also the notes in it. I had a book wherein I'd written all the formulae. There is one formula in notes of Power System. X = cube root of R. I remember there was a website that even had its derivation, but I don't have the bookmarks...

Homework Statement [/B] Use the Taylor remainder theorem to give an expression of ##\sqrt 2 - P_3(1)## P_3(x) - the degree 3 Taylor polynomial ##\sqrt {1+x}## in terms of c, where c is some number between 0 and 1 Find the maximum over the interval [0, 1] of the absolute value of the...

Dear Everybody, I am having trouble to determine the value of delta when c is strictly greater than 0. Here is the work: The Problem: Find the Limit or prove that the limit DNE. $\lim_{{x}\to{c}}\sqrt{x} for c\ge0$ Proof: Case I: if c>0. Let $\varepsilon>0$ Then there exists $\delta>0$...

Homework Statement Suppose we have: ## f(x) = x^2 - b ## ## b > 0 ## ## x_0 = b ## And a sequence is defined by: ## x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i) } ## prove ## \forall i \in N ( x_i > 0 ) ## Homework Equations The Attempt at a Solution a)Here I tried solving for ## x_1 ## as...

Hi all, I have done the question in two methods. The first method is done by rational root test and the second method is by modulo p (theorem is as attached). It seems that my answers for both methods do not tally. 1. Where have I done wrong in the attached for the methods? Which is the...

Homework Statement Prove that 2 is a primitive root of ##\mathbb{Z}/83\mathbb{Z}## by hand. Hint: Think hard about ##2^{41}##. Homework Equations Euler's theorem, Euler's Totient function, Chinese remainder theorem(not sure if its relevant). We don't really have anything else. The Attempt at...

Homework Statement An oscillator when undamped has a time period T0, while its time period when damped. Suppose after n oscillations the amplitude of the damped oscillator drops to 1/e of its original value (value at t = 0). (a) Assuming that n is a large number, show that...

Suppose, that $f(x)=ax^2+bx+c$, where $a$,$b$ and $c$ are positive real numbers. Show, that for all non-negative real numbers $x_1,x_2,…,x_{1024}$ \[\sqrt[1024]{f(x_1)\cdot f(x_2)\cdot \cdot \cdot f(x_{1024})} \geq f\left ( \sqrt[1024]{x_1\cdot x_2\cdot \cdot \cdot x_{1024}} \right )\]

Dear all, Please solve this integral: I tried integral by substitution, but failed. Wolframalpha shows the result is 6, but I don't know how to proceed it. Can it be solved by elementary function?

Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { } (This question is probably asked a lot but I could not find it so I'll just ask it myself.) Does the square root of negative numbers exist in the complex field? In other words is...

We have some type of squash plant growing in a small garden. It remarkable how much it has spread. I would think it advantageous for such a plant to form additional root systems, try as I might I do not see them on this plant. Is that typical of squash and pumpkin type plants, no additional root...

I am reading Abbot's "Understanding Analysis," and in this text he assumes that the real numbers are complete, that is, he assumes the least upper bound property, and begins to prove everything from there. Later in the book he proves that the square root of 2 does in fact exist in...

Homework Statement What value of ##p## gives a double root for ##f(x;p) = \cos x - 0.8+px^2##? I'm using python. Homework Equations Nothing comes to mind. The Attempt at a Solution I was thinking about choosing a window ##p\in[a,b]## such that ##a=0.3## yields 4 roots and ##b=0.4## yields 0...

Homework Statement [/B] It's been a couple of years since differential equations so I am hoping to find some guidance here. This is for numerical analysis. Any help would be much appreciated. Homework EquationsThe Attempt at a Solution

Hi All, I am having trouble installing Root Explorer on my phone. Anyone know of any other free/cheap app that would allow me to search through my phone's root directory? Or, is there some other way I can find a file within an app on my phone, given the very primitive default file manager in...

if a is a complex number then sqrt(a^2)=? Is it is similar to Real Number? Help me please

Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?

Homework Statement Consider the curves: y = x^2 from 1/2 to 2 and y = \sqrt{x} from 1/4 to 4. a. Explain why the lengths should be equal. b. Set up integrals (with respect to x) that give the arc lengths of the curve segments. Use a substitution to show that one integral can be...

Hi, I working on their text this equation did not make sense to me. From equation 1 it differentiate second term , I wonder how he got second term of equation 2. What I think is, what I wrote at the bottom

Quick question: In the proof that the square root of 2 is irrational, when we are arguing by contradiction, why are we allowed to assume that ##\displaystyle \frac{p}{q}## is in lowest terms? What if we assumed that they weren't in lowest terms, or what if we assumed that ##\operatorname{gcd}...

Homework Statement The problem relates to a proof of a previous statement, so I shall present it first: "Suppose P is a self-adjoint operator on an inner product space V and ##\langle P(u),u \rangle## ≥ 0 for every u ∈ V, prove P=T2 for some self-adjoint operator T. Because P is self-adjoint...