  We develop an \textit{a posteriori} error analysis for a numerical estimate
of the time at which a functional of the solution to a partial differential
equation (PDE) first achieves a threshold value on a given time interval. This
quantity of interest (QoI) differs from classical QoIs which are modeled as
bounded linear (or nonlinear) functionals {of the solution}. Taylor's theorem
and an adjoint-based \textit{a posteriori} analysis is used to derive
computable and accurate error estimates in the case of semi-linear parabolic
and hyperbolic PDEs. The accuracy of the error estimates is demonstrated
through numerical solutions of the one-dimensional heat equation and linearized
shallow water equations (SWE), representing parabolic and hyperbolic cases,
respectively.
