In what type of control mode is the output rate of change dependent on the magnitude of the input?

The correct choice is Integral control mode. In this mode, the output response is not only based on the current value of the error (the difference between the setpoint and the process variable) but also takes into account the cumulative sum of past errors over time. This cumulative effect means that the output rate of change is directly dependent on the magnitude of the input error. Essentially, as the input error remains, the integral action continues to increase the output to eliminate the error, demonstrating a clear relationship between the input's magnitude and the output rate of change.

This relationship is particularly essential in processes where a persistent error must be addressed to achieve stability and accuracy in system performance. In contrast, other control modes may focus predominantly on the current error without considering the accumulated effects, which does not make them suitable for this specific question regarding dependency on the magnitude of the input.