The PCR Project has a great article looking at US attempts to win hearts and minds in the Muslim world.
Using the specific example of American aid in Pakistan following the 2005 South Asia Earthquake, the graph they use illustrates a dramatic increase in the favorable opinion of the United States. These gains- rising from 23 to 46%- were as quickly lost as they were gained. Following a January 2006 predator strike in Pakistani territory, ratings dropped back down to 27%