Molten carbonate fuel cells are well suited for stationary power production and heat supply. In order to enhance service lifetime, hot spots, respectively, high temperature gradients inside the fuel cell have to be avoided. In conflict with that, there is the desire to achieve faster load changes while temperature gradients stay small. For the first time, optimal fast load changes have been computed numerically, including a parametric sensitivity analysis, based on a mathematical model of Heidebrecht. The mathematical model allows for the calculation of the dynamical behavior of molar fractions, molar flow densities, temperatures in gas phases, temperature in solid phase, cell voltage, and current density distribution. The dimensionless model is based on the description of physical phenomena. The numerical procedure is based on a method of line approach via spatial discretization and the solution of the resulting very large scale optimal control problem by a nonlinear programming approach.