**A fly is crawling along the top of the curve, from left to right, y = 7-x^2. A spider waits at the point (4,0). Find the distance between the spider and fly when they first see each other.**

**Answer:**

The spider sees the fly when a tangent line to the curve intersects (4,0). In order to find the equation of this line, we take the derivative the curve and plug in (4,0) to find the equation of the line that meets this description:

$y = (7-x^2)$

$dy/dx = -2x$

The equation of the line that is tangent to the curve and intersects (4,0) is:

$y=-2x+b$

$0=-2(4)+b$

$b=8$

Now we have two equations and two unknowns. We can solve to find out where the tangent line and the curve intersect.

$-2x+8 = 7-x^2 $

$x^2-2x=-1$

$x(x-2)=-1$

The only value that solves this is x=1

Therefore,

$y=-2(1)+8 = 6$

The spider and fly will first see each other at (1,6)