It's an interesting idea, but I'm not so sure about your price structure. I guess that you don't want to bill for every notification, but have you thought about billing for prevented no-shows (maybe on top of a base-price that includes x appointments)? You could still make the case that your Reminder pays for itself after the first no-show and make more money off of users that benefit more from your product.

Also: I don't know who the target audience for the base-price is. If you know you have 10 and only 10 appointments per month, this plan may be fine, but otherwise? Why not change it to a "1$ for every appointment"-Plan?