Autonomous vehicles (AVs) have already driven millions of miles on public roads, but even the simplest scenarios have not been certified for safety. Current methodologies for the verification of AV's decision and control systems attempt to divorce the lower level, short-term trajectory planning and trajectory tracking functions from the behavioral rules-based framework that governs mid-term actions. Such analysis is typically predicated on the discretization of the state space and has several limitations. First, it requires that a conservative buffer be added around obstacles such that many feasible plans are classified as unsafe. Second, the discretized controllers modeled in this analysis require several refinement steps before being implementable on an actual AV, and typically do not allow the specification of comfort-related properties on the trajectories. In contrast, consumer-ready AVs use motion planning algorithms that generate smooth trajectories. While viable algorithms exist for the generation of smooth trajectories originating from a single state, analysis should consider that the AV faces state estimation errors and disturbances. Third, verification is restricted to a discretized state space with fixed-size cells; this assumption can artificially limit the set of available trajectories if the discretization is too coarse. Conversely, too fine of a discretization renders the problem intractable for automated analysis. This work presents a new verification tool, APEX, which investigates the combined action of a behavioral planner and state lattice-based motion planner to guarantee a safe vehicle trajectory is chosen. In APEX, decisions made at the behavioral layer can be traced through to the spatio-temporal evolution of the AV and verified. Thus, there is no need to create abstractions of the AV's controllers, and aggressive trajectories required for evasive maneuvers can be accurately investigated.