General first-order unit-step response: for a stable, positive-gain first-order transfer function, describe how the output approaches its final steady state (qualitative trajectory shape).

Introduction / Context:
First-order systems appear everywhere: tanks, thermal masses, RC circuits. Their hallmark is an exponential approach to a new steady state following a step input, with trajectory governed by the time constant τ and steady-state gain K.

Given Data / Assumptions:

• Single real pole at s = −1/τ, τ > 0.
• Positive steady-state gain K > 0.
• Stable system (pole in left half plane).

Concept / Approach:
The standard step response is y(t) = K * (1 − e^(−t/τ)) for t ≥ 0 (assuming zero initial bias). This rises smoothly from 0 toward K without overshoot or oscillation. Its slope is maximum at t = 0 and decays exponentially; the output never decreases at any time when K > 0.

Step-by-Step Solution:

Verification / Alternative check:
At t = τ, the response reaches about 63.2% of final; at t ≈ 5τ, it is essentially at steady state, consistent with a monotonic rise.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing first-order behavior with second-order underdamped responses that can overshoot.

Final Answer:
A monotonic increase