In fluid mechanics and dynamics, the trajectory of a water jet from a nozzle or a projectile fired from a cannon is naturally described using parametric equations, where time $t$ is the parameter. The horizontal position is $x(t) = v_0 \cos(\theta) t$, and the vertical position is $y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$. By applying parametric differentiation $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, an engineer can determine the exact slope of the water jet at any specific horizontal distance without needing to algebraically eliminate time $t$ to form a complex Cartesian equation.

Highway engineers often use spiral curves to smoothly transition vehicles from a straight tangent section into a circular curve. These clothoid spirals are most easily modeled using polar coordinates $r = f(\theta)$. By calculating the angle $\psi$ between the radius vector and the tangent line ($\tan \psi = \frac{r}{r'}$), engineers can design the precise entry angle required for the pavement to safely guide vehicles into the curve without abrupt steering changes.

Find the slope of the tangent line to the parametric curve $x = t^2 + t$ and $y = t^3 - 3t$ at $t = 1$.

Find the second derivative $\frac{d^2y}{dx^2}$ for the curve defined by $x = t^2$ and $y = t^3$.

Find the points where the curve $x = 2t^3 + 3t^2$, $y = t^3 - 12t$ has horizontal or vertical tangents.

Find the slope of the tangent line to the cardioid $r = 1 + \sin \theta$ at $\theta = \frac{\pi}{3}$.

Find the points on the curve $r = 2 \cos \theta$ where the tangent line is horizontal.

Find the angle $\psi$ between the radius vector and the tangent line for the logarithmic spiral $r = 3e^{2\theta}$ at any angle $\theta$.

Find the angle $\psi$ for the circle $r = 5$ at $\theta = \frac{\pi}{6}$.