The functional expansion of the nonuniform excess Helmholtz free energy density functional around the bulk density was truncated at the first order, and then the functional counterpart of the Lagrangian theorem of differential calculus was employed to make the truncation formally exact. The concrete procedure is the following: According to the Lagrangian theorem and the definition of the direct correlation function (DCF), the original expansion coefficient, that is, the uniform first-order DCF, was replaced by the nonuniform first-order DCF whose density argument is an appropriate mixture of the density distribution and the bulk density with an adjustable parameter. With reference to the weighted density approximation, the nonuniform first-order DCF was then approximated by its uniform counterpart with a weighted density as its density argument. The weighting function was specified by equating the second-order functional derivative of the present truncated nonuniform excess Helmholtz free energy density functional with respect to the density distribution in the uniform limit to the second-order bulk DCF; the normalization condition of the weighting function specifies the adjustable parameter to be 0.5. The truncated expansion was incorporated into the density functional theory (DFT) formalism to predict the nonuniform hard-sphere fluid density distribution in good agreement with simulation data for three confining geometries: a single hard wall, a spherical cavity, and a bulk hard-sphere particle. The nonuniform Lennard-Jones fluid was investigated by dividing the bulk second-order DCF into a strongly density-depending short-range part and a weakly density-depending long-range part. The latter is treated by functional perturbation expansion truncated at the lowest order whose accuracy depends on how weakly the long-range part depends on the bulk density. The former is treated by the present truncated approximation. The two approximations are put into the density profile equation of the DFT to predict the density distribution for the Lennard-Jones fluid under the influence of two external potentials. The predicted density distribution displays even higher accuracy than that of two previous density functional perturbation theories in the region away from the repulsive wall but is a little inferior to the former approaches in the region near the wall. Also, a functional integral procedure was employed to develop the density functional approximation for the nonuniform first-order DCF to calculate the global thermodynamic properties of the nonuniform fluid.