Tagged Questions
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
0answers
77 views
Continuous, differentiable, continuously differentiable
I came across the following problem:
Let $\alpha \in \mathbb R$. Where is the function continuous, differentiable, continuously differentiable?
$$f(x) =
\begin{cases}
x|x|^\alpha & ...
0
votes
1answer
83 views
For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?
I came across the following question:
For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?
What I have tried:
Since for $ \alpha = 1 $ is clearly non-differentiable in ...
3
votes
3answers
107 views
How does one calculate the integral of the sum of two absolute values?
I know how to find the integral of just one absolute value, but this problem presents the integral of the sum of two absolute values. Help!
I want to evaluate:
$$ \int_a^b{(|x-1| + |x+1|) dx} $$
3
votes
3answers
110 views
Exposition On An Integral Of An Absolute Value Function
At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I ...
0
votes
1answer
75 views
Absolute function continuous implies function piecewise continuous?
I have a simple true/false question that I am not sure on how to prove it.
If $|f(x)|$ is continuous in $]a,b[$ then $f(x)$ is piecewise continuous in $]a,b[$
Anyone that can point me in the ...
0
votes
2answers
72 views
True/false question: limit of absolute function
I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit ...
1
vote
1answer
85 views
double integral of an absolute function
I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
2
votes
2answers
178 views
Why is the derivative of $\frac{|x|}{x}$ equal to $\emptyset$ at $x=0$?
I got a bit of a confusion here. If $\varphi(x)=\frac{|x|}{x}$, then
$$
\varphi(x) = \left.\Bigg\{
\begin{array}{cc}
1 &if \ x>0\\
\emptyset & if \ x=0\\
-1 & if \ x <0
\end{array} ...
5
votes
1answer
666 views
Derivatives of functions involving absolute value
I noticed that if the absolute value definition $\lvert{x}\rvert=\sqrt{x^2}$ is used then we can get derivatives of functions with absolute value, without having to redefine them as piece-wise.
For ...
2
votes
2answers
271 views
Spivak Calculus 3rd ed. $|a + b| \leq |a| + |b|$
I'm working through the first chapter of Michael Spivak's Calculus 3rd ed.
Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ ...
3
votes
6answers
297 views
Evaluating $\int |x|^3 \; dx $
$$\int |x|^3 \; dx $$
In my module it is suggest to use integration by parts,
$$ \text{ Set }I = \int (|x|^3 \cdot 1) \; dx = |x|^3 \cdot x - \int \color{red}{\frac {x^3}{|x|^3}3x^2}\cdot x \; dx$$
...
1
vote
1answer
171 views
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 $x=a.$
1
vote
1answer
8k views
Integral of an absolute value function
How do I find the definite integral of an absolute value function?
For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
1
vote
2answers
108 views
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:
$$
\begin{cases}\sqrt{x^2 ...
2
votes
2answers
79 views
How do I find this limit?
How would i find the limit as $\lim\limits_{x\to3}\frac{4x(x-3)}{|x-3|}$? that is the absolute value of x-3 in the denominator. I thought my professor told my class that we were able to omit the ...
3
votes
1answer
174 views
Is the book wrong about this left-hand limit with absolute value? (But, my delta depends on x.)
The book says that $$\lim_{x \rightarrow 0^{-}} \left( \frac{1}{x} - \frac{1}{|x|} \right) \mbox{does not exist}$$
But, given any $M \lt 0$ of large magnitude, if I choose $\delta = \frac{-x^{2}M}{2}$ ...
