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Push the angle towards $\pi/2$
In the Angular–Hyperbolic Lab, fix $b = 1$ and decrease $t$ towards $0$ . Record how $\mathcal{L}\{x^n \sin bx\}(t)$ behaves for $n = 0, 1, 2$ . Which values stay finite, and why?
A strong contribution: Reports observations and explains them using $\sin((n+1)\theta)/r^{n+1}$ with $r \to 1$ .
- Lab: Angular–Hyperbolic Lab
Foundation
Watching two roots collide
In the Root Geometry Lab, move $c$ in $x^2 - 2x + c$ from $0$ to $2$ . Describe the path of the roots and what happens exactly at $c = 1$ .
A strong contribution: Roots move along the real axis, meet at $x = 1$ , then split vertically into a conjugate pair.
- Lab: Root Geometry Lab

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