search for a section of log that was appended upon execution of script

I'm using WebLogic start script startmymanageserver1.sh that writes(appends) to a managedserver.log log file every time it is executed. The managedserver.log may have logs from the past few months which I cannot override. I wish to grep for a string Server is in RUNNING State not from the entire managedserver.log; but only from the section of logs that was appended as a result of my executing the startmymanageserver1.sh Can you please suggest how to grep from only the section of managedserver.log that was updated as a result of my last run of startmymanageserver1.sh?