This paper considers the optimization of static and dynamic appointments for a sequence of customers with random processing times served by a single agent. In the static case, the problem is to find the appointment times for the customers that minimize the expected weighted sum of idle time of the agent, waiting time of the customers, and overtime of the agent. The dynamic formulation is original. It limits cascading delays by using warning times, which are defined relative to the start of a previous job, to tell customers when to arrive for their job. The problem in this formulation is to find the warning times which minimize the aforementioned objective. We outline an algorithmic framework based on infinitesimal perturbation analysis (IPA) and stochastic gradient descent (SGD) that can be used to solve both problems. For the static case, we show that the SGD method converges to a global minimizer with no assumptions on the joint distribution of the processing durations. For the dynamic case, we construct a warning time schedule which provably outperforms the static scheme under certain conditions. We give empirical evidence of the efficacy of both approaches using both synthetic data as well as data collected from a year of elective surgeries at a local hospital. Our results suggest the validity of the dynamic formulation as a more cost-efficient solution than the static formulation to the appointment scheduling problem.