Drone Flight Path Linearization (Robotics & Computer Engineering)

An autonomous inspection quadcopter follows a non-linear parabolic search altitude trajectory $y(x)$ (measured in meters) as it surveys a construction site along a horizontal path $x \in [0, 5]$:

The drone's embedded real-time micro-controller guidance system cannot execute heavy non-linear optimization algorithms fast enough during rapid aerodynamic gust disturbances. Instead, the autopilot computes the tangent line equation (linearization) at the current navigation waypoint $x = 3$ to approximate the path locally.

1. Using the difference quotient limit definition, derive the general derivative function $y'(x)$ for the flight path.
2. Determine the quadcopter's exact altitude $y(3)$ and instantaneous flight slope $m = y'(3)$ at waypoint $x = 3$.
3. Construct the tangent line linear approximation equation $L(x)$ at $x = 3$.
4. Use $L(x)$ to estimate the drone's altitude at $x = 3.1\text{ m}$, and compare the linear estimate with the true non-linear trajectory value $y(3.1)$.