1- You have studied the graphs of rational functions and in particular asymptotes.

In photography fixed focal length means that the focal length is not adjustable. Photographers are unable to zoom in and out on a particular subject when using such a lens. Not being able to easily zoom in on a subject might seem like a huge disadvantage. But such lenses are credited with being able to produce much higher quality pictures in controlled settings.

In order for a film camera with a fixed focal length F to focus on an object located a distance x from the lens, the film must be placed a distance y behind the lens. F, y, and x are related as follows

Now suppose a camera has a lens with focal length F = 65.

a) Explain what happens to the focusing distance y as the object moves far away from the lens.

b) Explain what happens to the focusing distance y as the object moves closer and closer to the lens.

c) In general, why is it not possible to cross a vertical asymptote?

2- Discuss the intermediate value theorem. For the intermediate value theorem why do you think it is necessary for the signs of f(a) and f(b) to be different in order to guarantee there is a zero between a and b.

If it is the case that the signs of f(a) and f(b) are the same, does that mean the function f(x) cannot have a zero between a and b? Explain.