Question

Find the slope of the curve at the indicated point.
f(x)=|x|
a.) x=2
b.) x=-3

Answer 1

For \(x>0\), you have \(|x|=x\). For \(x<0\), you have \(|x|=-x\).

So at x=2, \(f'(x)=\dfrac{d}{dx}[x]\bigg|_{x=2}\)

And at x=-3, \(f'(x)=\dfrac{d}{dx}[-x]\bigg|_{x=-3}\)

Answer 2

what is the d/dx
I don't think i have learned about that

Answer 3

That's the derivative operator. \(f'(x)=\dfrac{d}{dx}[f(x)]\).

Is this for calculus?

Answer 4

yea but i barely started and havent got to that point

Answer 5

Well it's the exact same thing as what you already (presumably) have been taught. If you're more comfortable with \(f'\) notation, I'll stick to that.

Since \(f'(x)\) gives you the slope of the tangent line to \(f(x)\) at some value of \(x\), all you have to do is find the derivative of \(f(x)\), then plug in the desired value.

So, like I mentioned before, you have
\[f(x)=\begin{cases}x&\text{for }x\ge0\\-x&\text{for }x<0\end{cases}\]
The first value you're given is 2, which is positive, so you use the first piece of the function; \(f(x)=x\).

Find \(f'(x)\), then plug in 2. In other words, find \(f'(2)\).

Answer 6

oh so that's is it? Thank you!

Answer 7

Yep. Just keep in mind that part (b) has \(f(x)=-x\).