We use impedance spectroscopy to measure the high-frequency properties of single-walled carbon nanotube field effect transistors (swCN-FETs). Furthermore, we extend scanning gate microscopy (SGM) to frequencies up to 15 MHz and use it to image changes in the impedance of swCN-FET circuits induced by the SGM tip gate. In contrast to earlier reports, the results of both experiments are consistent with a simple RC parallel circuit model of the swCN-FET, with a time constant of 0.3 micros. We also use the SGM tip to show the local nature of the memory effect normally observed in swCN-FETs, implying that nanotube-based memory cells can be miniaturized to dimensions of the order of tens of nm.