1. Matrix Transformations & Invariant Lines | Shear & Stretch Effects – Deriving the transformation matrix, finding the invariant line, and determining conditions for area scaling.

2. Mathematical Induction & Differentiation | Proof for General Polynomial Form – Proving a formula involving derivatives of tangent functions using induction.

3. Polynomial Equations & Root Relations | Quartic Transformations – Constructing new quartic equations from given roots and summing their powers.

4. Summation & Series Manipulation | Method of Differences & Closed-Form Expressions – Evaluating finite sums using cancellation techniques.

5. Polar Coordinates & Curve Properties | Conversion & Area Computation – Transforming Cartesian to polar form, sketching curves, and finding enclosed areas.

6. Rational Functions & Asymptotic Behavior | Stationary Points & Graphing – Identifying asymptotes, extrema, and sketching rational curves.

7. Planes & Perpendicular Distances | Intersection & Angular Separation – Finding plane equations, angles between planes, and shortest perpendicular distances between skew lines.