The flow in a single-cavity extrusion die is considered by using 1-D equations. Since the fluid flow in a well-performing die does not deviate much from the case of perfect widthwise distribution, linearization of 1-D governing equations is often justified. It is demonstrated how linearization can lead to significant simplifications in the analysis of die flow. A self-consistent asymptotic expansion is identified to accomplish this linearization; the expansion can be generalized to obtain higher-order corrections if necessary. In contrast to previous analyses, which have generally been numerical, analytic solutions are obtained to predict widthwise flow nonuniformities in a die having a specific cavity area variation and in which cavity inertial effects are important; these solutions are valid for power-law and Newtonian fluids. A novel truncated power-law approach to modeling the flow of shear-thinning fluids in a die is proposed that combines, by using analytical criteria, the solutions for purely Newtonian or power-law flow in the cavity and slot. The validity of this approach is demonstrated through comparison with numerical results of a 1-D die model for a Carreau constitutive equation.