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Using the density-matrix renormalization-group (DMRG) technique, we calculate critical exponents for the one-dimensional Hubbard model with open boundary conditions with and without additional boundary potentials at both ends. A direct comparison with open boundary condition Bethe ansatz calculations provides a good check for the DMRG calculations on large system sizes. On the other hand, the DMRG calculations provide an independent check of the predictions of conformal field theory, which are needed to obtain the critical exponents from the Bethe ansatz. From the Bethe ansatz we predict the behavior of the 1/L-corrected mean value of the Friedel oscillations (for the density and the magnetization) and the characteristic wave vectors, and show numerically that these conjectures are fulfilled with and without boundary potentials. The quality of the numerical results allows us to determine the behavior of the coefficients of the Friedel oscillations as a function of the Hubbard interaction.