1. let f(x) = 2/x2 . you would like to use this function as the pdf over a range [a, b]. you set a = 1. find the value of b such that both the the above properties for pdf hold. solution

freeborn women in ancient rome were citizens (cives), but could not vote or hold political office. because of their limited public role, women are named less frequently than men by roman historians. but while roman women held no direct political power, those from wealthy or powerful families could and did exert influence through private negotiations. exceptional women who left an undeniable mark on history range from lucretia and claudia quinta, whose stories took on mythic significance; fierce republican-era women such as cornelia, mother of the gracchi, and fulvia, who commanded an army and issued coins bearing her image; women of the julio-claudian dynasty, most prominently livia (58 bc-ad 29), who contributed to the formation of imperial mores; and the empress helena (c.250–330 ad), a driving force in promoting christianity.

women in the ancient greek world had few rights in comparison to male citizens. unable to vote, own land, or inherit, a woman’s place was in the home and her purpose in life was the rearing of children. this, though, is a general description, and when considering the role of women in ancient greece one should remember that information regarding specific city-states is often lacking, is almost always from male authors, and only in athens can their status and role be described in any great detail. neither are we sure of the practical and everyday application of the rules and laws that have survived from antiquity. we do know that spartan women were treated somewhat differently than in other states. for example, they had to do physical training like men, were permitted to own land, and could drink wine.

the factors that the economies of the muslim empires prosper were:

1- land trade routes-located between three continents  

2- sea trade routes

trade contributed to the spread of muslim empires, and with that islamic culture led to a growing feeling of internationalism resulted in the transfer of technology, science, and other cultural forms.

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