∆ Difference Quotient

The 5-step dance — write, expand, subtract, factor, take the limit — applied to polynomial and rational functions. The bridge from secant slopes to tangent slopes. Every problem is solved the long way; the power rule is a future shortcut, not a current tool.

Problems

• Q1 — Tangent slope of f(x) = 2x² + 1 (the 5-step dance)
• Q2 — General difference quotient (stop one step earlier)
• Q3 — Tangent line at a specific point (5 steps at x = 2)
• Q4 — Secant slope of f(x) = 2/x (sign trap on f(-1))
• Q5 — General difference quotient of f(x) = −1/x (two sign traps)