Tagged Questions

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 & ...
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 ...
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 ...
2
votes
2answers
179 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 $x^2-4=0$ ...
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} ...
5
votes
1answer
654 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 ...
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
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2answers
158 views

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 ...
0
votes
1answer
344 views

derivative of absolute value of a complex function

If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get ...