All questions with tag [math: absolute-value]


How to manage the absolute value on a differential equation $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$

Hi every person I require to address a formula of this type: $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$ with borders problems. The absolute value is my trouble. Certainly without it, the remedy of these is well dealt with in the literary works. After search in the inquiries I located this: So, can I do the very same procedure? Or there is an additio...
2022-07-25 20:41:51

Two Analysis Questions

1) Define : $\langle z\rangle := (1+|z|^2 ) ^\frac{1}{2} $ for all $z \in \mathbb{C} $. Prove : $\langle x+y\rangle \leq 2\langle x\rangle\langle y\rangle $ for all $x,y \in \mathbb{R} ^N$ . 2) Define: $ S^\beta := \left\ \leq c_n \langle x\rangle^\beta -n \right\ $ for some $c_n<\infty$, all $x \in \mathbbR $ and all $ 0\leq n \in ...
2022-07-25 15:57:04

Finding the absolute extreme values for a multivariable function

Find the outright severe values taken by $f(x,y) = x^2 + 4y^2 + x - 2y$ on the shut area confined by the ellipse $1\over4$$x^2 + y^2 = 1$. I recognize this could be a standard inquiry yet could a person please clarify just how to address this trouble? Many thanks beforehand.
2022-07-24 09:23:44

Inequality absolute value

If $x>0,y>0$, why does it adhere to for any kind of $y \in (x,x+1)$, that from: $$|f(y)-f(x)|<1,$$ we have $$|f(y)| \leq |f(x)|+1$$
2022-07-24 09:18:26

Evaluating $\int |x|^3 \; dx $

$$\int |x|^3 \; dx $$ In my component it is recommend to make use of integration by components, $$ \text{ Set }I = \int (|x|^3 \cdot 1) \; dx = |x|^3 \cdot x - \int \color{red}{\frac {x^3}{|x|^3}3x^2}\cdot x \; dx$$ $$ \implies I = |x|^3 \cdot x - \color{red}{3\int |x|^3\;} dx$$ $$ \implies I = \frac 1 4 |x|^3 \cdot x +C $$ I am having probl...
2022-07-22 18:37:36

Integral of an absolute value

I am attempting to review the indispensable $$\int_{-1}^2 (x-2|x|)\,dx$$ I recognize that this need to offer me $x^2 /2 - x^2$ for the antiderivative. I after that review at $2$ which offers me $2 - 4 = -2$ Then review at $-1$ and also get $\frac{1}{2} - 1 = -\frac{1}{2}$. After that I locate the distinction $-2 - \frac{1}{2} = -2.5$ This i...
2022-07-22 15:31:14

The relationship between the derivative of $f(x)$ and $|f(x)|$

I have seen it in an exercise book. I don't know how to do it. If $f(x)$ is continuous at $x=a,$ and $|f(x)|$ can be differentiated at $x=a,$ then $f(x)$ is differentiable at $x=a.$
2022-07-20 00:50:50

I can't find a absolute value function that have [-1,1] range

I desire a function $f:\mathbb{R}\to[-1,1]$ with absolute value like $f(x)=|a-x|\ldots$ that have $[-1,1]$ array. Can any person aid me?
2022-07-17 17:39:54

Math Database For Problem Descriptions In An App.

I am creating an application for children and also they will certainly have a selection of troubles from percent troubles, absolute value troubles, adverse number troubles, portion troubles, etc I was wanting to have a summary of just how to do each type prior to they started. Does any person recognize of a good data source where I could get all...
2022-07-16 17:25:35

Prove that $||x|-|y||\le |x-y|$

I've seen the complete evidence of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I have not seen the evidence of the reverse triangular inequality: \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please confirm this making use of just the Triangle Inequality over? Thanks significantly.
2022-07-16 17:10:41

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ps. At some point the equations become so stiff that s becomes negative and if I improve tolerances of the numerical integrator, it crashes. To solve this problem, I'd like to approximate the ODEs with s_new=abs(s) and...
2022-07-16 16:54:43

Integral of an absolute value function

How do I locate the precise indispensable of an absolute value function? As an example: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
2022-07-14 02:19:53

The solution set of the equation $|2x - 3| = - (2x - 3)$

The solution set of the equation $\left | 2x-3 \right | = -(2x-3)$ is $A)$ $0$ , $\frac{3}{2}$ $B)$ The empty set $C)$ (-$\infty$ , $\frac{3}{2}$] $D)$ [$\frac{3}{2}$, $\infty$ ) $E)$ All real numbers The correct answer is $C$ my solution: $\ 2x-3 = -(2x-3)$ when $2x-3$ $\geqslant$ $0$ $\Rightarrow$ $x$ = $\frac{3}{2}$ $-(2x-3) = -(2x-3)$ wh...
2022-07-14 01:24:34

The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm)

Suppose we have actually a set $S$ of actual numbers. Show that $$\sum_{s\in S}|s-x| $$ is marginal if $x$ amounts to the median. This is an example test inquiry of among the tests that I require to take and also I do not recognize just how to continue.
2022-07-10 04:54:04

Does the triangle inequality for the absolute value hold for matrix trace?

It is well - recognized that, $\left|m-n\right|\ge\left|\left|m\right|-\left|n\right|\right|$ genuine numbers. Yet if one specifies $\left|M\right|=\sqrt{M^2}$ for a symmetrical matrix $M$, does one have $$\operatorname{trace}\left(\left|M-N\right|\right)\ge\operatorname{trace}\left(\left|\left|M\right|-\left|N\right|\right|\right)$$ if $M\ne\le...
2022-07-07 02:32:19

When is possible to ignore an absolute value

$e^{\frac{2}{x-1}\log\left|x-1\right|}+1\neq 0$ Since that this is an exponential function, this equation is verified $\forall x \in \mathbb{R}$? Or I have to consider the absolute value of the ligarithm? Note: these are not homework.
2022-07-06 23:35:12

How to manipulate absolute values when shifting parts in an inequality

I have the adhering to inequality $|4x - 2| \le 0.5$ I intend to adjust this so it is simply $|x|$ on one side, and also every little thing else on the various other, yet I'm not exactly sure just how the absolute value makes complex points. Just how would certainly I deal with this?
2022-07-06 00:52:32

Fields with their own absolute value

Let $F\hspace{.02 in}$ be a field. $\:$ Let $E\hspace{.02 in}$ be a non-zero subring of $F$. Let $\hspace{.03 in}\leq\hspace{.03 in}$ be a total order on $E\hspace{.02 in}$ that makes $E\hspace{.02 in}$ an ordered ring. $\;\;\;$ Let $\;\; |\cdot| \: : \: F\to E \;\;$ be a function. I will require the following to hold for all members $x$ of $E...
2022-07-06 00:36:30

How to set up the existence condition of an absolute value

$$ \frac{\sqrt{4 + \arccos\left|\frac{2-x}{x+3}\right|}}{\sqrt{x^2 - 4x + 5} - 3} $$ I'm trying to find the natural domain of the function above. I set up this conditions: $$ \begin{cases}\sqrt{x^2 - 4x + 5} - 3\neq0&(denominator)\\x^2 - 4x + 5\ge0&(root)\\4 + \arccos\left|\frac{2-x}{x+3}\right|\ge0&(root)\\\left|\frac...
2022-07-06 00:13:05

How to minimize an equation with absolute values?

How would certainly I deal with decreasing the expression $\left(|z_1| + |z_2|\right) \times \left(|z_1 + z_2| + |z_1 - z_2|\right)$ based on the restraint $|z_1|^2 + |z_2|^2 = 1$ considered that $z_1$ and also $z_2$ can be intricate numbers? I considered attempting Lagrange multipliers, yet it does not appear feasible due to the fact that ...
2022-07-04 20:34:31