No, batteries are consumables, and are not covered for three years even if you have AppleCare. If a battery is defective, the defect will nearly always show up during the first year and need to be addressed then. Defective batteries are dealt with by Apple on a case-by-case basis, considering their ages, cycle counts, charge capacities and any physical deformations. As I wrote above, it's extremely unlikely that a three-year-old battery will be replaced for free, AppleCare or no.