You already have a task manager. Press and hold the home button and it will open. From there you can stop apps or background processes either individually or in bulk.
To see what is using up the battery go to "settings" and then "Battery". There you can see what app/processes have used how much of the battery since the last recharge. Note that the battery stats are reset every time the phone is recharged.
It is possible after a major update to the OS that the battery may appear to discharge faster than normal. After a few charge cycles all then appears normal again. Apparently some battery management systems needs a few cycles to charge or indicate correctly again.