For questions about or involving the absolute value function.

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5
votes
6answers
165 views

Prove that $|-x| = |x|$

Using only the definition of Absolute Value: $\left|x\right| = \begin{cases} x & x> 0 \\ -x & x < 0 \\ 0 & x = 0,\end{cases}$ Prove that $|-x| = |x|.$ This seems so simple, but I ...
3
votes
2answers
119 views

Derivative of $x\cdot|x|$ on $x=0$?

$$f(x) = x |x|$$ Wolfram Alpha says is: $$f'(x) = \frac{2x^2}{|x|}$$ and thus $f'(0)$ is indeterminate, while an HP48 says that: $$f'(x) = |x| + x \operatorname{sgn} x,$$ which would yield $f'(0) ...
1
vote
1answer
47 views

Integrating an integrand with an absolute value on exponential

This is one heck of an embarrassment but it is amazing how these bits of subtlety gets lost in the back of the head after the first year of undergraduate studies-with every computation chucked into ...
2
votes
3answers
154 views

Find all $z$ such that $\left|\tan z\right| = 1$

Find all z such that $$\left|\tan z\right| = 1$$ The first thing that came to my mind was to write tangent in terms of $e^z$ and take its modulus, but I couldn't solve it in this way.
0
votes
1answer
32 views

Normal distribution question involving absolute value

I don't quite understand how to do the following question. How I tried to do it is to imagine the normal distribution curve, with the highest peak at 4. I understand that |Q| means the absolute value ...
3
votes
2answers
62 views

Is finding the second derivative of $\sqrt[3]{\vert x\vert}$ the best method to determine if it is convex?

I have an exercise where I have to tell on which intervals a function is concave or convex. I usually do it using second derivative, but I would like to know if there is a simpler way of doing so, ...
0
votes
5answers
96 views

Is the absolute value of zero positive or negative?

If I had $|x|$, then we know, for pretty much any $x$, that the following is true:$$|x|\ge0$$$$|0|=0?$$Which, by the nature of how we usually apply the absolute value, the solution is positive and ...
0
votes
2answers
32 views

One absolute value inside of another absolute value in the equation

Let's say we have an equation: $||x|-2| = |2|x|+4|$ How does one go solving it? Symbolab says that it currently doesn't support step by step explanation for this problem, so I would really ...
2
votes
2answers
36 views

Why is there $(2,3)$ corner coordinates on the graph, using this function: $f(x)=2x-1+|x-2|?$

Currently I am studying about absolute values and I had to consider this function and draw it's graph: $$f(x)=2x-1+|x-2|$$ I helped myself with symbolab, here is the link: https://www.symbolab.com/...
1
vote
1answer
44 views

How Does This Line Intersection Equation Work?

For the lines: $y = ax + b$ $y = cx + d$ The standard intersection equation $x_i = \frac{d - b}{a - c}$ $y_i = \frac{ad - bc}{a - c}$ If the points that I was given to find the line $y = ax + ...
1
vote
3answers
81 views

Integrating $f(x)=\int|\cos(x)|dx$ and then solving $f(x)=\frac {2x}{\pi}$?

I realised the other day that by applying absolute value signs to the cosine function and then integrating, I would get an almost sine function that doesn't have negative slope. And then I also ...
0
votes
1answer
26 views

Validity of inequalities using integrals and absolute value

This question is similar to this one but the only response was pointing out mistakes in the solution. My goal is to determine whether the operator $T: C[0,1] \to C[0,1]$ defined by $Tx = \int_{0}^{t}...
0
votes
1answer
36 views

Integrating absolute terms

This is just to clarify my doubt regarding absolute values functions. Lets say there is a function $$f(x) = ax^{2} - \left|\frac{bx}{c}\right|$$ and we are asked to integrate this over $-\infty \to \...
3
votes
2answers
44 views

Absolute value equation with rational expression

I am to solve the equation: $|\frac{2x}{x^2 - 3} | < 1$ And so: 1. I rewrote it as $|2x| < |(x - \sqrt 3) | |(x + \sqrt 3)|$ And I tried to divide it into a few intervals For $ x\in(-\infty;...
1
vote
1answer
67 views

how to minimize a summation containing an absolute value

Thank you all in advance I have been having some trouble figure out the following problem. You are given a sample {$y_i$}, i=1,… N, from an unknown probability distribution p(y). I want to show the ...
1
vote
2answers
36 views

Integration of $|e^{-(2+j)t}|^2$

The integration of $|e^{-(2+j)t}|^2$ from zero to infinity is $1/4$ when I separate above as $|e^{-2t}|^2 \cdot |e^{-2jt}|^2$ and integrate. $|e^{-2jt}|$ was taken as $1$. But when I integrate the ...
5
votes
4answers
151 views

Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents?

I understand that $\sqrt{x^2} = |x|$ because the principal square root is positive. But since $\sqrt x = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of ...
5
votes
1answer
190 views

Verify integration of $ \int\frac{\sqrt{2-x-x^2}}{x^2}dx $

This is exercise 6.25.40 from Tom Apostol's Calculus I. I would like to ask someone to verify my solution, the result I got differs from the one provided in the book. Evaluate the following integral: ...
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
197 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
48 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
628 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
57 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 $\lvert f(x) \rvert - \lvert L \rvert <= \lvert f(x) - L \rvert < \epsilon$...
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 ...
2
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 $s,t$ $$ |\varphi(s)-\...
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
27 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 $2)$...
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
44 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
133 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
50 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
61 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
83 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
69 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
113 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 ...