SEC2_GYM · Calculus · CALC.DERIV.APPLICATIONS

Curve analysis and optimisation

Intuition

Understand first, compress later.

Calculus studies change and accumulation through limiting processes.

Switch representations

Secant becomes tangentf(x) = x², x = 1

curve fsecant Δx = 1.2secant Δx = 0.6secant Δx = 0.2tangent, slope 2point of tangency

All three secants pass through (1, 1). As Δx shrinks (1.2 → 0.6 → 0.2) the secant slope moves 3.2 → 2.6 → 2.2 — towards the tangent slope f′(1) = 2. That limit is the derivative.

Formal view

You can curve analysis and optimisation when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.