Highway Exit Ramp Grade Design (Construction Engineering)

A construction management site engineer oversees earthwork and grading for an elevated highway interchange off-ramp. The vertical road elevation profile $H(x)$ (in meters above sea level) over the station interval $x \in [0, 4]$ (measured in hundreds of meters) follows the rational descent curve:

To meet AASHTO roadway safety standards, construction grading crews must verify that the road slope (grade) at the merge station $x = 2$ matches the connecting collector road grade of $-1.50\text{ m/station}$ ($-1.5\%$) and assemble the tangent grade line equation for surveying stakes.

1. Using the difference quotient limit definition, find the instantaneous grade function $H'(x) = \frac{dH}{dx}$.
2. Calculate the exact slope of the off-ramp at station $x = 2$ and determine if it complies with the $-1.50$ roadway grade specification.
3. Formulate the linear tangent grade equation $y(x)$ at $x = 2$ to place surveyor reference grade stakes along the embankment.