We study properties of W01,p(R+,tβ) — the completion of C0∞(R+) in the power-weighted Sobolev spaces W1,p(R+,tβ), where β∈R. Among other results, we obtain the analytic characterization of W01,p(R+,tβ) for all β∈R. Our analysis is based on the precise study of the two trace operators: Tr0(u)≔limt→0u(t) and Tr∞(u)≔limt→∞u(t), which leads to the analysis of the asymptotic behavior of functions from W01,p(R+,tβ) near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems in ODEs, or PDEs with the radial symmetries. We can also apply our results to some questions in the complex interpolation theory, raised by Cwikel and Einav (2019), which we discuss within the particular case of Sobolev spaces W1,p(R+,tβ).