# difference quotient and absolute value

• Sep 26th 2007, 01:21 PM
sinewave85
difference quotient and absolute value
"Evaluate the difference quotient for the given functions. Show all of your work."

The ones I am stuck on are:

a. f(x) = lxl if x<-1 and 0<h<1

b. f(x) = lxl if x>1 and 0<h<1

I know the answers -- a. -1, b. 1 -- but I got them by plugging numbers that fit the parameters into the difference quotient and then evaluating it, which would be fine if I didn't have to show my work. As far as evaluating it directly, thought, I have no idea what to do. (My latex skills are pathetic, so please excuse my not typing all of my work out.)
• Sep 26th 2007, 01:34 PM
topsquark
Quote:

Originally Posted by sinewave85
"Evaluate the difference quotient for the given functions. Show all of your work."

The ones I am stuck on are:

a. f(x) = lxl if x<-1 and 0<h<1

b. f(x) = lxl if x>1 and 0<h<1

I know the answers -- a. -1, b. 1 -- but I got them by plugging numbers that fit the parameters into the difference quotient and then evaluating it, which would be fine if I didn't have to show my work. As far as evaluating it directly, thought, I have no idea what to do. (My latex skills are pathetic, so please excuse my not typing all of my work out.)

Let's do a):

$\displaystyle \frac{|x + h| - |x|}{h}$

Now, x is less than -1 and 0 < h < 1 we know that x + h is negative. Thus $\displaystyle |x + h| = -(x + h)$. Again, since x is negative $\displaystyle |x| = -x$. Thus
$\displaystyle \frac{|x + h| - |x|}{h} = \frac{-(x + h) - (-x)}{h} = \frac{-x - h + x}{h} = \frac{-h}{h} = -1$

-Dan
• Sep 26th 2007, 02:17 PM
sinewave85
Quote:

Originally Posted by topsquark
Now, x is less than -1 and 0 < h < 1 we know that x + h is negative. Thus $\displaystyle |x + h| = -(x + h)$. Again, since x is negative $\displaystyle |x| = -x$. Thus

$\displaystyle \frac{|x + h| - |x|}{h} = \frac{-(x + h) - (-x)}{h} = \frac{-x - h + x}{h} = \frac{-h}{h} = -1$

-Dan

:eek: Oh, wow. I never really understood absolute value correctly untill just now. That is both amazing and sad. Oh, well. Glad I decided to do calculus, or I would have gone my whole life not understanding absolute value. Just sad. Thanks for the enlightenment.