HCI Tutorial Applications of Differentiation Q8

In optics, the focal length of a thin lens (negligible thickness) is the distance between the centre of the lens and the point at which all rays will meet after passing through the lens. The focal length is a fixed quantity for each piece of lens. The distances from the object \(A\) to the lens and from the lens to the image \(B\), for a lens of negligible thickness, in air, are related by the thin lens formula \(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\), where \(u\) is the distance of an object from the lens, \(v\) is the distance of the image from the lens, and \(f\) the focal length of the lens. Suppose \(A\) can vary with time \(t\) (the object can move closer to or further away from the lens).

1. Show that the further \(A\) is from the lens, the nearer \(B\) is to the lens.[2]
2. If \(A\) is moving away from the lens at a rate of \(2\) cm/s when it is \(45\)cm from the lens, find the rate of change of distance of \(B\) in terms of \(f\).[4]