For questions about or involving the absolute value function.

learn more… | top users | synonyms

1
vote
2answers
24 views

Find the points at which $f$ has an absolute maximum or minimum on $I$ without graphing

Assume $I=[0.9,3.1], f:I\rightarrow\mathbb{R}$ is defined by $f(x):=|x^2-4x+3|, x\in I$. Without sketching the graph of $f$ on $I$, find points at which $f$ has an absolute minimum on $I$ and points ...
1
vote
4answers
196 views

What is the name for this operation?

What is the name for this operation? Effectively, take a range and adjust its center point to 0 on a number line. In the case of the example above, I'm specifically looking for the name or ...
1
vote
1answer
16 views

Some equations involving multiple absolute values

Consider the following equation: $$|x+y^2|+|x-y^2|+|y+x^2|+|y-x^2|=a$$ I'm looking for the method for solving some problems regarding this equation, namely: 1) prove that if $a=2015$, then the ...
1
vote
1answer
34 views

Silly question about complex numbers - if its modulus is < 1, does raising it to higher exponents make it decrease to the real number 0?

Just playing around with the modulus definition doesn't really confirm that thought... Is it true? If |z| = |x+iy| < 1, is $$\lim_{n\to \infty} z^n = 0 ?$$ Thanks,
0
votes
4answers
47 views

What exactly is this equation?

Thank you for assistance, I'm just having issues remembering what this is called? For example, the equation would go like this |x+1| = 4 What is this type of equation called, with the two | | ? ...
0
votes
1answer
17 views

Find extreme values of absolute function

I have to find the extreme values of the following function: $f(x) = |x-2|+|x+3|$ on [-5;5]. How do I do that?
7
votes
2answers
625 views

A ''strange'' integral from WolframAlpha

I want integrate: $$ \int \frac{1}{\sqrt{|x|}} \, dx $$ so I divide for two cases $$ x>0 \Rightarrow \int \frac{1}{\sqrt{x}} \, dx= 2\sqrt{x}+c $$ $$ x<0 \Rightarrow \int \frac{1}{\sqrt{-x}} \, ...
0
votes
1answer
52 views

Double absolute value inside integral

Any ideas as to go about doing this particular integral? $$\int\limits\limits_{-1}^{4}||x^2+x-6|-6| dx$$ I'm a bit confused as to how to consider the cases into account. My idea was to consider 4 ...
1
vote
2answers
55 views

If $\lim_{x \to x_0} f(x) = L$, then $\lim_{x \to x_0} \lvert f(x)\rvert = \lvert L \rvert$.

If $\lim_{x \to x0} f(x) = L$, then $\lim_{x \to x0} \lvert f(x)\rvert = \lvert L \rvert$. I know this is true, because $\lvert f(x) \rvert - \lvert L \rvert <= \lvert f(x) - L \rvert < ...
0
votes
1answer
34 views

complex absolute value equations

How on earth does one solve this problem? I know that to solve abs value equations we have to consider both the negative and postive possibilities, but the constant (6) makes this a little more ...
3
votes
2answers
53 views

Property of function $\varphi(x)=|x|$ on $\mathbb{R}$

Define $\varphi(x)=|x|$ on $[-1,1]$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $\varphi(x+2)=\varphi(x).$ How do you prove that for any $s,t$ $$ ...
0
votes
0answers
29 views

Exploring n + abs(x)

In class we are focussing on 'Lattice Theory' at the moment. I have an assignment with instructions saying simply - "Explore the following:" followed by a list of 5 theorems. The purpose of the ...
-1
votes
1answer
73 views

Find the absolute and relative error for a calculator with incorrect rounding

A calculator is out of order. The calculator will round up every single number to the nearest integer if the value at the first decimal digit is 6 and above, or else it rounds down the number to be ...
1
vote
3answers
27 views

Absolute value of a complex number with a arbitrary basis

I want to calculate the square of the absolute value of a complex number $x^{ia}$, with $x$ and $a$ being real while $i$ is the imaginary number: $$\left|x^{ia}\right|^2=?.$$ I have trouble because ...
0
votes
1answer
26 views

Function with absolute value and parameters?

I need help with this exercise Consider $f(x)=||x|-a|$ $1)$Determine as $a\in \mathbb{R}$ varies the intervals in which $f(x)$ is continuous and the intervals in which $f(x)$ is differentiable ...
0
votes
1answer
35 views

Expressing maximum and minimum as $\frac12(x+y)\pm\frac12|x-y|$

I'm looking at ways to get the max/min value of two numbers without using conditional statements, I found these two functions: ...
0
votes
2answers
22 views

How would you solve an inequality in the form: $|f(x)| < g(x)$?

An inequality such as $|x + 1| < x + 3$ was given to be solved. I attempted to used the theorem: $|x + c| < \delta \implies c-\delta < x < c+\delta$ But $x \in \Bbb{R}$ so $x$ could be ...
2
votes
2answers
43 views

Initial values problem with absolute value

I've some doubts about initial values problems involving differential equation with absolute values. For example if I have a differential equation like $y'=|x+1|$ with initial condition $y(3)=-2$, ...
2
votes
0answers
131 views

Summation of the absolute value of the variable

The summation of cosine $\sum_{k=1}^N \cos (k x)$ is well known (for example, see the previous question here) and is called Lagrange's trigonometric identity. Is it possible to construct a similar ...
2
votes
2answers
64 views

Prove that $|ab+1|>|a+b|$ with $|a|<1$, $|b|<1$

Prove that $|ab+1|>|a+b|$ with $|a|<1$, $|b|<1$ $a$, $b$ are real numbers Where $|a|$ is the absolute value of $a$. Every time, I arrive to a dead-end.
1
vote
2answers
49 views

Indefinite integrals with absolute values

Which is the right way to solve indefinite integrals which contain absolute values? For example if I have $\int |2x+3| e^x dx$ Can I consider the sign function and integrate separetly? I mean doing: ...
0
votes
1answer
28 views

Prove $|1+z^{2n}|\geq1-|z|^{2n}$

i need to prove that $|1+z^{2n}|\geq1-|z|^{2n}$, I have tried use that $|a-b|\geq||a|-|b||$ but I think that it is not so, help?, $z\in D(0,1)$
0
votes
4answers
104 views

Why isn't the derivative of $|2x^2-3x|$ equal to $|4x-3|$?

I don't quite understand why this is the case? Since when differentiating $|2x^2-3x|$ you get $\frac{(2x^2-3x)(4x-3)}{|2x^2-3x|}$...... when it is $2x^2-3x$, the derivative is $4x-3$ and when it is ...
0
votes
0answers
55 views

How to determine whether expression is positive or negative?

Given expressions $|x - 3 + y|$ and $|x + 3 + y|$ how can I determine, whether are those positive or negative, and determine their value in the intervals of: $y < -x - 3$ $y \in [-x - 3, 3 - x)$ ...
0
votes
2answers
23 views

Is $ \left| \lim_{t \rightarrow 0}{\frac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \frac{|f(x+ty) - f(x)|}{|t|}}$

Suppose $X$ is a Banach space and $f:X \rightarrow \mathbb{R}$ is a continuous function. Is it true for all $x,y \in X$ that $$ \left| \lim_{t \rightarrow 0}{\dfrac{f(x+ty) - f(x)}{t}} \right| \leq ...
1
vote
2answers
23 views

Absolute value inequality verification

This is a pretty trivial question, but I'm trying to list out steps to show that if $|x-c|<1\Rightarrow |x|\leq |c|+1$. Is there a trick with the triangle inequality? Thanks
0
votes
1answer
69 views

Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 8t + 8 cot(t/2), [π/4, 7π/4]

Having a little bit of trouble figuring out this problem here. Find the absolute maximum and absolute minimum values of $f$ on the given interval. $f(t)$ = $8t + 8 cot(t/2)$, ...
2
votes
2answers
39 views

How to integrate $|x^n|$

How to integrate $|x^n|$? The answer was given as $\frac{|x^n|x}{n+1}$. How do you find this? I would also like to know how do find the anti derivative of a modulo function when interval is not given. ...
1
vote
4answers
76 views

Why if $a<b$ and $-a<b$ we can say that $|a|<b$?

Why if $a<b$ and $-a<b$, then we can say that $|a|<b$? Maybe this is trivial by I don't know how to proof it.
0
votes
2answers
68 views

How can we prove that $\sqrt{ x^{2} }$ is equals to $|x|$?

I used to use this equality at school. But now in my books of Analysis this property is not mentioned. Is this maybe incorrect?
2
votes
0answers
46 views

Reformulate absolute value as quadratic problem

I'm looking for standard approach to reformulate this objective function. The aim is to find values of $x_i$ that are close to either $y_i$ or $-y_i$ ($y_i$s are known) in a least-squares sense: ...
1
vote
1answer
104 views

Expected value of the absolute value of the difference of two random variables

I have to compute the absolute value of an estimator defined as $T_5=\frac{1}{2}E[|X_1-X_2|]$ in order to state if it is unbiased for $\sigma$, where $X$ is distributed as a $N(0,\sigma^2)$. I am ...
2
votes
1answer
50 views

Ideas for a limit calculation

The limit to show is the following: $$ \lim_{t\to \infty} \int_\mathbb{R}\left|\frac{-\sin x \sin tx}{x^2} \right|dx $$ A direct splitting of the integral into $\int_{-\infty}^0+\int_0^{+\infty}$ in ...
0
votes
1answer
19 views

Question about solving absolute value equation.

What is the sum of all possible solutions to this equations? $|x+4|^2 -10|x+4|=24$ My attempt: Since $(x+4)^2=|x+4|^2$, so I can ignore the absolute sign of the first term. So we only need to deal ...
4
votes
5answers
93 views

Solve $ \left|\frac{x}{x+2}\right|\leq 2 $

I am experiencing a little confusion in answering a problem on Absolute Value inequalities which I just started learning. This is the problem: Solve: $$ \left|\frac{x}{x+2}\right|\leq 2 $$ The answer ...
0
votes
1answer
29 views

Inequality Involving Absolute Value

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If the derivative $f'$ of $f$ is also continuous on $[0,1]$, prove that: $ \int_0^1 |f'(x)-f(x)| dx \geq \frac{1}{e}$. ...
1
vote
1answer
36 views

How to expand this expression?

How to express this expression $\frac{z}{2}<|y|<z$. Is it correct to expand it as following $-z<-\frac{z}{2}<y<\frac{z}{2}<z$
1
vote
2answers
43 views

When solving an equation with absolute value on both sides, how to choose the side to work with?

When solving an equation with absolute value on both sides, such as $$|2x-1|=|4x+3|$$ how to choose one side of which to use the definition of absolute value? For example, if we apply absolute value ...
0
votes
1answer
37 views

How to find the minimum of $c|1+x|^n+|1-x|^n$

How to find the minimum of the \begin{align} f(x)=c|1+x|^n+|1-x|^n \end{align} for $n \ge 1$ and $c > 0$. If we take the derivative of $f(x)$ we get \begin{align} f'(x)=-c {\rm sign}(1+x) ...
0
votes
0answers
31 views

Cauchy Determinant with Absolute Values

This is perhaps a straightforward question but I'm a little confused. An $n\times n$ Cauchy matrix $A$ is a matrix with entries $$a_{i,j}=\frac{1}{x_i-y_j}$$ for $1\le i,j\le n$, where $x_i$ and $y_j$ ...
0
votes
1answer
69 views

if |f| is periodic then f is periodic [duplicate]

Decide whether the following statement about a function f: R -> R is true. If |f| is periodic, then f is periodic. Give a proof or counterexample.
0
votes
2answers
68 views

How do I solve a quadratic inequality with absolute value using cases?

$$\left| x^{ 2 }-5x+5 \right| \le x$$ Steps I took: Using the quadratic formula, I split the solutions up into: Case 1: $$ x^{ 2 }-5x+5\le 0$$ $$x\le \frac { 5+\sqrt { 5 } }{ 2 } and\quad x\ge ...
0
votes
1answer
96 views

Prove or disprove: If $ f $ is periodic, then $|f|$ is also periodic.

I started with the definition of periodic function and absolute value function. And I do it with discussing different cases of $x$ and $p$. But I got stuck with when $ -p\leq x <0 $ , I want to ...
1
vote
1answer
32 views

Meaning of abs. value of a sigma field?

In a hand out I saw the a notation which looks to be the absolute value of a sigma field F, |F|. I googled it but I could not really find what it means and the notation confuses me. Anyone that might ...
1
vote
1answer
30 views

Can one realize the real part of every entire function $f$ as $\ln| g|$ with $g$ entire?

Let $\Re$ denote real part and $|\cdot|$ absolute value. Does there exist, for every entire $f$, an entire $g$ such that $\Re f = \ln |g|$ ?
0
votes
0answers
56 views

Use axioms to solve inequality $| x-2| +| x-4| < 1$

I have a feeling that the inequality is false for all values of x, but I don't know at which point that should have become clear to me. I am supposed to solve the inequality using $x < a$ ...
3
votes
2answers
35 views

Solve $[\frac{x^2-x+1}{2}]=\frac{x-1}{3}$

How can one solve the equation : $[\frac{x^2-x+1}{2}]=\frac{x-1}{3}$ ? Such that $[x]$ is the integer part of $x$. By definition : ...
3
votes
5answers
98 views

How to solve $\lvert{x}\rvert - \lvert{2+x}\rvert = x$?

How do I solve the following equation? $$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$ I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way ...
1
vote
2answers
41 views

Integral of absolute value: $\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$

I am stuck trying to integrate $$\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$$ Incidentally, I'm interested in solving equation (5) in this paper using the Laplace distribution. I just got ...
2
votes
2answers
59 views

Solution of integral involving exponential and absolute values [closed]

I want to solve this integral $\int_0 ^\infty e^{-iwx}e^{-α|x|} dx$ Any ideas on how to solve it?