Macbook not discovering Apple TV for Airplay after Mavericks update
I think I found the problem, my Macbook is mid- 2010. Mavericks multiple-displays through Airplay only works for Macbook early 2011 or newer, as Apple states in this page


  Multiple Display with Apple TV and an HDTV works with Apple TV (2nd generation or newer), iMac (mid 2011 or newer), Mac mini (mid 2011 or newer), MacBook Air (mid 2011 or newer), MacBook Pro (early 2011 or newer), and Mac Pro (late 2013).


However, an app named AirParrot works for older systems. 

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