Wireless and wearable ultrasound devices promise to enable continuous ultrasound monitoring, but power consumption and data throughput remain critical challenges. Reducing the number of transmit events per second directly impacts both. We propose a task-based adaptive transmit beamforming method, formulated as a Bayesian active perception problem, that adaptively chooses where to scan in order to gain information about downstream quantitative measurements, avoiding redundant transmit events. Our proposed Task-Based Information Gain (TBIG) strategy applies to any differentiable downstream task function. When applied to recovering ventricular dimensions from echocardiograms, TBIG recovers accurate results using fewer than 2% of scan lines typically used, showing potential for large reductions in the power usage and data rates necessary for monitoring. Code is available at https://github.com/tue-bmd/task-based-ulsa.