In a 3 1/2- or 4-quart slow cooker combine carrots, sweet pepper, and onion. Sprinkle with tapioca. Place pork roast on top of vegetables. Pour barbecue sauce over meat. Cover and cook on low-heat setting for 6 to 7 hours or on high-heat setting for 3 to 3 1/2 hours.
Remove meat from slow cooker, reserving juices. Thinly slice meat. Return sliced meat to slow cooker; stir to coat with sauce. Serve meat on hamburger buns.