# Thread: How do I find the slope of a tangent line that crosses two different parabolas?

1. ## How do I find the slope of a tangent line that crosses two different parabolas?

Okay, so I have this one problem on my Calculus homework that I'm struggling with. I've thought about it all day long, but I'm not getting anywhere. Here's the problem word for word:

"Two small arches have the shape of parabolas. The first is given by f(x)=1-x^2 for x= [-1, 1] and the second by g(x)= 4 - (x - 4)^2 for x=[2, 6]. A board is placed on top of these arches so it rests on both. What is the slope of the board? Hint: Find the tangent line to y=f(x) that intersects y=g(x) in exactly one point."

I don't really want someone to answer it for me, or I'll never learn. I just need a good shove (a better shove that the "hint" in the book).

2. ## Re: How do I find the slope of a tangent line that crosses two different parabolas?

Have you tried the hint? How do you get the slope of the tangent lines to f(x)?

3. ## Re: How do I find the slope of a tangent line that crosses two different parabolas?

The slope of f(x) = f'(x) and the slope of g(x)=g'(x).

And...

f'(x)= -2x
g'(x)= -2x+8

That's about as far as I've been able to get all day. I've experimented with some things, like giving the points some generic names like:

P(z,f(z))
Q(s,g(s))

And then setting up the change in y over change in x formula for slope and setting it equal to one of the derivatives, but that was just a shot in the dark and didn't get me far. I'm not really sure what to do.

4. ## Re: How do I find the slope of a tangent line that crosses two different parabolas?

$\text{Hint: If } (x_0,y_0) \text{ is the point at which the tangent line touches } f(x), \text{ then equation of the tangent line is given by } y=(-2x_0)x+b.$
This is the starting point, and I'll check back on this later.