We have performed first-principles calculation of the surface and bulk wavefunctions of the Cu(111) surface and their hybridization energies to a Co adatom, including the potential scattering from the Co. By analyzing the calculated hybridization energies, we found the bulk states dominate the contribution to the Kondo temperature, in agreement with recent experiments. Furthermore, we also calculate the tunneling conductance of a scanning tunneling microscope (STM) and compare our results with recent experiments of Co impurities in the Cu(111) surface. Good quantitative agreement is found at short parallel impurity-tip distances (< 6 A). Our results indicate the need for a new formulation of the problem at larger distances.