Tapping-mode atomic force microscopy (AFM), in which the vibrating tip periodically approaches, interacts and retracts from the sample surface, is the most common AFM imaging method. The tip experiences attractive and repulsive forces that depend on the chemical and mechanical properties of the sample, yet conventional AFM tips are limited in their ability to resolve these time-varying forces. We have created a specially designed cantilever tip that allows these interaction forces to be measured with good (sub-microsecond) temporal resolution and material properties to be determined and mapped in detail with nanoscale spatial resolution. Mechanical measurements based on these force waveforms are provided at a rate of 4 kHz. The forces and contact areas encountered in these measurements are orders of magnitude smaller than conventional indentation and AFM-based indentation techniques that typically provide data rates around 1 Hz. We use this tool to quantify and map nanomechanical changes in a binary polymer blend in the vicinity of its glass transition.