Optimality criteria for the selection of a response surface design have been presented by many authors. For the most part these criteria have led to optimal designs for predicting response. In many applications of response surface methodology, however, the primary experimental objective can best be fulfilled by estimating the slope of the response surface. This would be the case, for example, in preliminary experiments for exploring the factor space. Criteria for selecting a design to estimate the slope have been presented only for certain situations. In this dissertation the integrated mean squared error criterion for the selection of a response surface design has been generalized. The generalized criterion applies to the selection of an experimental design to estimate specified linear functions of the parameters of a polynomial model in k≥1 independent variables using the method of least squares. When the linear parametric functions are chosen to be the first order partial derivatives of the fitted polynomial, the criterion leads to optimal designs for estimating the slope. It is found that the generalized criterion function can be minimized in the situation in which the slope is to be estimated using the partial derivatives of a fitted equation of first order. The optimal designs are two-level designs augmented in some cases by center points. They are obtainable using 2^k-p fractional factorial designs and the Plackett and Burman two-level designs. For the case in which the slope is to be estimated using the partial derivatives of a fitted equation of second order, the generalized criterion function cannot be minimized. Because of this the criterion function is expressed as the sum of a bias component and a variance component. An attempt is then made to obtain optimal designs with respect to each component. When the variance component is considered alone, moment conditions are obtained that characterize the optimal designs. Then conjectured optimal designs are obtained for the situation in which there are two independent variables and the region of experimentation is of finite extent. This is done for both the case in which the region of interest is a hypercube and the case in which it is a hypersphere. For the bias component it is found that the optimal designs are rotatable designs regardless of whether the region of interest is a hypercube or a hypersphere. Because the optimal designs are not uniquely determined with respect to the design moments, minimization of the variance component within the class of optimal designs with respect to the bias component is used as a secondary design criterion. This leads to uniquely determined designs with respect to the design moments and methods are given for constructing the designs. / Ph. D.