This article presents a hybrid distributed-parameters model and an adaptive control framework for microcantilevers utilised in atomic force microscope systems for controlled force manipulations. The model assumes a general nonlinear interaction force between the microcantilever's tip and the surface of the sample. This interaction force includes the sample's surface and probe's tip distance as well as the first and second derivatives of this force implicitly. Despite such detailed modelling of interaction force, there are a number of uncertainties including tip mass, damping coefficients and nature of the interaction force that would affect the response of the system and hence, an adaptive controller is needed to compensate for these unmodelled dynamics and uncertainties. Unlike the current practices that deal with the lumped-parameters model of the cantilever, a comprehensive distributed-parameters model based on the Euler-Bernoulli theory is considered here. An adaptive controller is then designed such that by giving a force input to the base of the microcantilever, the tip of the microcantilever can track a desired trajectory despite the flexibility of the microcantilever and aforementioned uncertainties. Extensive simulation results are provided to illustrate that the microcantilever's tip can asymptotically follow a harmonic trajectory even for a system with higher modes of vibration when it is designed based on single-mode model.