2
votes
3answers
178 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
0
votes
2answers
70 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
0
votes
1answer
43 views

How do I expand absolute values?

If we have this expression: $$f = uu-\left( u + \frac{\partial u}{\partial x} \delta x \right) \left( u + \frac{\partial u}{\partial x} \delta x \right)$$ we can expand it to this: $$f = u^2-\left( ...
0
votes
2answers
69 views

Different ways of defining Absolute Value

Calculus I presents this definition of absolute value: $$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$ But you can also ...
4
votes
2answers
351 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 $a$ ...
0
votes
2answers
86 views

Two Analysis Questions

1) Define : $\langle z\rangle := (1+|z|^2 ) ^\frac{1}{2} $ for all $z \in \mathbb{C} $. Prove : $\langle x+y\rangle \leq 2\langle x\rangle\langle y\rangle $ for all $x,y \in \mathbb{R} ^N$ . 2) ...