0

So with this problem I am completely lost on where to even start this problem to get to the answer that my professor wants me to prove. The questions is shown in the attached picture.

Let f(x) = abs(x), a) Show that f'(x) = 2*theta(x) - 1

I honestly would just like some help on where to start/ point me in the right direction with both part a in problems G and H. Ive looked everwhere online and cant find anything that would be useful. Ive even tried using the notes from class and there are no similar problems that were even gone over. Thanks so much!

J Greiner
  • 3
  • 2
  • Can you tell us what $\Theta(x)$ and $\delta (x)$ are? – Matti P. Sep 11 '20 at 09:55
  • So for what we have from class is that $$ \theta $$ as a piecewise function. So when x < 0 its 0 and when x > 0 it will be 1. The $$ \delta $$ from looking at previous examples in class with it used is the dirac delta. – J Greiner Sep 11 '20 at 09:59
  • The $$\theta$$ is called the heaviside step function – J Greiner Sep 11 '20 at 10:06

1 Answers1

0

By definition we have $$|x|=\begin{cases} x \space\space\space\space\space\space\text{if $x\geq 0$}\\ -x \space\space\space\text{if $x< 0$}\end{cases}$$

Thus $$f'(x)=\begin{cases} 1 \space\space\space\space\space\space\text{if $x> 0$}\\ -1 \space\space\space\text{if $x< 0$}\end{cases}$$ $$=2\theta(x)-1$$ where $$\theta(x)=\begin{cases} 1 \space\space\space\text{if $x> 0$}\\ 0 \space\space\space\text{if $x< 0$}\end{cases}$$

Next note that the derivative of the Heaviside's unit step function is the Dirac delta function.

You want to show that $f'(x)=2\theta(x)-1$, from the definition above we have $$f'(x)=\begin{cases} 1 \space\space\space\space\space\space\text{if $x> 0$}\\ -1 \space\space\space\text{if $x< 0$}\end{cases}$$

Do you agree with this?

Since we have defined $$\theta(x)=\begin{cases} 1 \space\space\space\text{if $x> 0$}\\ 0 \space\space\space\text{if $x< 0$}\end{cases}$$

then what can we do to $\theta(x)$ so that it gives $f'(x)$?

Well:

$$2\theta(x)=\begin{cases} 2 \space\space\space\text{if $x> 0$}\\ 0 \space\space\space\text{if $x< 0$}\end{cases}$$ so $$2\theta(x)-1=\begin{cases} 1 \space\space\space\text{if $x> 0$}\\ -1 \space\space\text{if $x< 0$}\end{cases}$$

which is $f'(x)$.

Alessio K
  • 10,599
  • Im still a little lost on where i would need to start with this problem. Am i trying to show that f'(x) will lead to the dirac delta, then somehow turn it into the theta? – J Greiner Sep 11 '20 at 10:36
  • I've added more above, let me know if it's fine. – Alessio K Sep 11 '20 at 10:44
  • Awesome Thanks Ares, that actually help a lot, i though about doing something like that but i second guessed myself because that seemed to simple. – J Greiner Sep 11 '20 at 11:00