A manufacturer of lightbulbs wants to produce bulbs that last about .

some bulbs burn out faster than others.

Let  be the fraction of the companys bulbs that burn out before hours.

 always lies between  and .

 and , where  is large value.

(a)

Graph the :

Graph:

.

(b) Find the meaning of the derivative .

The rate at which the fraction  increases as  increases.

(c) The value of :

.

; all bulbs burn out eventually.

(a)

Graph:

Graph the :

.

(b) The rate at which the fraction  increases as  increases.

(c) .

All bulbs burn out eventually.