Unfortunately, using the update_time column from INFORMATION_SCHEMA.TABLES only works for MyISAM tables. It does not work for InnoDB.
The most effective way to get the last time a table was written is to rely on the OS. You must check the timestamp of the .ibd or .MYD files within a database folder.
I wrote earlier posts about how to do this:
Dec 21, 2011: Fastest way to check if InnoDB table has changedFastest way to check if InnoDB table has changedApr 04, 2013: How to check which tables in DB (MYSQL) updated in last 1 hour / last 1 minute?How to check which tables in DB (MYSQL) updated in last 1 hour / last 1 minute?Jun 03, 2013: Is there a way to find the least recently used tables in a schema?Is there a way to find the least recently used tables in a schema?Sep 25, 2014: want to find out which databases are used in last 30 days or notwant to find out which databases are used in last 30 days or notJan 16, 2015: How can I determine which Innodb table is being written?How can I determine which Innodb table is being written?