The absolute-value tag has no wiki summary.
2
votes
1answer
32 views
Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$
How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$?
This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
0
votes
0answers
36 views
Calculation of the sub gradient of the first norm of a matrix
Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
0
votes
1answer
19 views
How to graph an absolute value equation?
How would you graph:
$|x+y|=1$ ?
I can do the normal $y=|x+1|$ and all that. But how would you do a question with two of these unknowns in the absolute value?
Any help would be greatly appreciated, ...
1
vote
1answer
36 views
Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?
Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ .
Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ?
NB: The answer ...
0
votes
1answer
19 views
How to linearize the following LP
I want to minimize $|d_1-d_2|+e1+e2+e3$ where $d_1,d_2,e_1,e_2,e_3>=0$ and $|.|$ denotes the absolute value, for some linear constraints. Is there any way I can linearize the objective function?
5
votes
1answer
59 views
Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$
in my math class we were given a list of indefinite integrals, and one of them was:
$$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$
My working:
$$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int ...
1
vote
1answer
20 views
how to find absolute value for complex fraction
I have a Fourier transfer equation $H(jw) = \frac{jwL}{(jw)^2LC+jw\frac{L}{R}+1}$, and I need to find frequency to make $|H(jw)|$ is max.
I know I should take the derivative of $|H(jw)|$ then find ...
0
votes
2answers
50 views
Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs (Proved Differently) [duplicate]
My Proof:
This problem has mainly four cases, they are as follows:
1) $a, b > 0$
2) $a, b < 0$
3) $a > 0 > b$
4) $a < 0 < b $
Let suppose that the sum of the real numbers $a ...
1
vote
1answer
41 views
What is the modulus of a number?
What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb ...
0
votes
5answers
76 views
Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs
My Proof:
$|a +b| = |a| +|b|$ ..... $(i)$
$|a +b| < |a| + |b|$ ..... $(i)$
If $'a'$ and $'b'$ have same signs:
Let $a$ and $b$ be equal to $-x$. Replacing $a$ and $b$ with $-x$ in the equation ...
1
vote
2answers
72 views
Absolute Value of $|-3 -2|$
$|-3 -2|$ is the distance between the points $-3$ and $-2$. If we solve it further then,
in one way I get $|-5| = 5$. But $5$ is the distance between $0$ and $-5$ in this case. In other way,
$2 ...
0
votes
2answers
33 views
Absolute Value Problem, Solution and Method
Please check my method and also if I have solved the following problem correctly:
Problem: $f(x) = |x - \frac12| + |x + \frac12|$
If $x = -1$, then:
$f(-1) = |-1 - \frac12| + |-1 + \frac12|$
From ...
2
votes
3answers
36 views
Question about absolute value in inequalities
My book presents the following: $$7 \le x \le 9 $$ so $$ -1 \le x - 8 \le 1 $$ and $$ |x-8| \le 1$$
I usually get confused with the way that taking the absolute value of an expression works. Could ...
0
votes
3answers
41 views
finding values for absolute convergence
Find all values of real number p or which the series converges:
$$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$
I tried using the root test and the ratio test, but I got stuck on ...
0
votes
1answer
43 views
Absolute of a trig function
Consider the function $$f(x) = 1\dfrac{1}{2} - 3\sin \left(\dfrac{1}{2}x \right). $$
I need to find the absolute of this function, which to my eye would just be
$$ f(x) = 1\dfrac{1}{2} + 3\sin ...
2
votes
2answers
43 views
Absolute Convergence of a Series
Find all values of real number p for which the series converges absolutely
$$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
0
votes
2answers
74 views
Calculating the integral $ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$
How do we calculate the following integral:
$$ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$
1
vote
2answers
86 views
How to evaluate the inequality $|x+1|<-1$?
Okay perhaps the title isn't specific enough, I didn't know how to word it exactly. I'm finding the interval of convergence for a power series and i know the answer to be (-2,0]
I end up with the ...
1
vote
2answers
43 views
Can summations distribute across absolute values?
Can I distribute a summation as follows?
$$
k\sum_{x \in X} \left| x - b \right| = \left| \left(k\sum_{x \in X}x \right) - \left( k\sum_{x \in X}b \right) \right|
$$
0
votes
0answers
76 views
Continuous, differentiable, continuously differentiable
I came across the following problem:
Let $\alpha \in \mathbb R$. Where is the function continuous, differentiable, continuously differentiable?
$$f(x) =
\begin{cases}
x|x|^\alpha & ...
1
vote
2answers
43 views
Finding absolute max and min values of function
Function given as $f(x,y) = 3x^2 + 2xy^2$. If $(x,y)$ lies in the region inside including edges of the triangle in the first quadrant given by $x\ge0, y\ge0, y\le2-x$. Reduce $f$ to a single variable ...
0
votes
1answer
82 views
For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?
I came across the following question:
For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?
What I have tried:
Since for $ \alpha = 1 $ is clearly non-differentiable in ...
0
votes
1answer
26 views
Is this (or when) does this equality hold?
Let $a,b,c,d \in \mathbb{R}$ and $x,y$ are variables which are also real numbers
$$|ax + by|^2 + |cx + dy|^2 + 2|ax + by||cx + dy| = (ax + by)^2 + (cx + dy)^2 + 2(ax + by)(cx + dy)$$
Is this always ...
2
votes
2answers
59 views
proving $|x - 1| < {1\over4} \Rightarrow |2x - 1| \geq {1\over 2}$
I tried solving the above, consider that: ($x \in R)$,
I know it's not a complicated problem to solve though I struggle getting on with this question,
What I've done far is:
$|x-1|<{1\over4} ...
1
vote
3answers
55 views
Find $x$ for absolute value inequalities
I'm trying to figure out this inequality:
$|x+1| + |x| \leq x^2$
I thought about trying it with two cases: $ (x = -x)$ and $(x = +x)$
but I don't seem to find out how to go through from here,
...
0
votes
1answer
54 views
Finding The Contour Maps Of A Function Of Two Variables
I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps.
Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$
I know I ...
1
vote
3answers
67 views
Solving $2|x+1|>|x+4|$
I'm trying to solve the following equations and inequalities for $x\in\mathbb R$:
$$2|x+1|>|x+4|$$
I know I'm supposed to consider the intervals $(-\infty,-4), [-4,-1]$ and $(-1,\infty)$ but ...
-1
votes
3answers
103 views
Determine all solutions to $|x+12|+|x-5|=15$
Determine all solutions to the following.
$$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$
1
vote
1answer
147 views
Finding the points of the curve where the tangent line is horizontal
The curve given is $\displaystyle y = \ln|x-2| + x + \frac{12}{x-2}$.
Find the points of the curve where the tangent line is horizontal.
My first stumbling block is the absolute value function. I ...
3
votes
3answers
105 views
How does one calculate the integral of the sum of two absolute values?
I know how to find the integral of just one absolute value, but this problem presents the integral of the sum of two absolute values. Help!
I want to evaluate:
$$ \int_a^b{(|x-1| + |x+1|) dx} $$
4
votes
2answers
55 views
Prove $||a| - |b|| \leq |a - b|$ [duplicate]
I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got:
$$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$
Subtracting $|b|$ from both sides yields,
...
5
votes
2answers
86 views
Question regarding usage of absolute value within natural log in solution of differential equation
The problem from the book.
$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$
I understand the solution till this part.
$\ln \vert 6 - y \vert = x + C$
The solution in the book is $6 - Ce^{-x}$
...
3
votes
3answers
108 views
Exposition On An Integral Of An Absolute Value Function
At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I ...
1
vote
4answers
170 views
Truth set of $-|x| \lt 2$?
An exercise in my Algebra I book (Pearson and Allen, 1970, p. 261) asks for the graph of the truth set for $-\left|x\right| \lt 2; x \in \mathbb{R}$.
I've re-stated the inequality in the equivalent ...
2
votes
3answers
180 views
Proof of triangle inequality
I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
2
votes
2answers
177 views
How to find critical points of an absolute values function
I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have?
I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
-1
votes
1answer
38 views
Integral of a mode squared [closed]
Hi could someone show me how to calculate $\psi_0$ out of the equation below?
$$\int \limits^{}_{V} \big|\psi_0 \sin (\omega t - kx) \big|^2 \, \textrm{d} V = 1\\$$
1
vote
2answers
43 views
Absolute ratios
I'm curious about the following idea:
suppose we have two values $P$ and $Q$, and the magnitude of the ratio $\frac{P}{Q}$ is between $0$ and $\infty$. If $P$ is smaller, then it's between $0$ and ...
1
vote
3answers
106 views
Adding equations in Triangle Inequality Proof
Inequality to prove:
$|a+b|\leq |a| + |b|$
Proof:
$-|a| \leq a \leq |a|$
$-|b| \leq b \leq |b|$
Add 1 and 2 together to get:
$-(|a|+|b|)\leq a+b\leq|a|+|b|$
$|a+b|\leq|a|+|b|$
What is the ...
1
vote
1answer
57 views
Integration Involving the Absolute Function
How do I integrate the double integral of the form $|x^2-y|$ with the boundaries $-1\leq x\leq 1$ and $-1\leq y\leq 1$?
1
vote
2answers
85 views
Solving $| ax + b | \gt c$
Does $| a x + b | > c$ always result in two solutions, $x \gt \dfrac{c - b}{a}$, and $x \lt\dfrac{-c - b}{a}$?
If I understand correctly, the first solution, $x > \dfrac{c - b}{a}$, is only ...
1
vote
1answer
36 views
minimum of absolute value
If we consider the following problem
$$
\mathbb{E}[(Y-y)^2 | X=x]
$$
I can easily show that the minimum with respect to $y$ occurs at
$$
y=\mathbb{E}[Y |X=x]
$$
How can I find the minimum of
$$
...
0
votes
3answers
85 views
Graphing Absolute Value Functions
Given: $y = -|2x + 1|-3$
I came up with the graph of...
$1, -7$
$0, -5$
$-1, -3$
$-2, -1$
$-3, 1$
If you were to graph this, it would turn out to be an entirely straight line. This is an ...
5
votes
2answers
68 views
Is there a lower-bound version of the triangle inequality for more than two terms?
The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$
Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
1
vote
2answers
80 views
Log laws and modulus
If you have the log of a modulus, (like after integration), how do the log laws work?
So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
0
votes
1answer
74 views
Absolute function continuous implies function piecewise continuous?
I have a simple true/false question that I am not sure on how to prove it.
If $|f(x)|$ is continuous in $]a,b[$ then $f(x)$ is piecewise continuous in $]a,b[$
Anyone that can point me in the ...
0
votes
2answers
68 views
True/false question: limit of absolute function
I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit ...
1
vote
1answer
83 views
double integral of an absolute function
I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
2
votes
1answer
59 views
Separable first-order linear equation and absolute value removal
We can use the integral of $\frac{1}{x}$ in order to solve a separable first-order linear equation like this:
$\frac{dy}{dt} + f(t) y = 0$
$
ln |y| = \left(-\int f(t)\,dt\right) + C
$
and then:
...
2
votes
2answers
177 views
Why is the derivative of $\frac{|x|}{x}$ equal to $\emptyset$ at $x=0$?
I got a bit of a confusion here. If $\varphi(x)=\frac{|x|}{x}$, then
$$
\varphi(x) = \left.\Bigg\{
\begin{array}{cc}
1 &if \ x>0\\
\emptyset & if \ x=0\\
-1 & if \ x <0
\end{array} ...
