Kap 2 wrap — mixed apps & light DE

V1 — Related-rate sketch from max/min Tools.

Builds on: f′ as slope Tool + chain (trig/product) used in apps. Not a brand-new related-rates chapter dump.

Tool gained: light related-rate read — differentiate a job relation F(x,y)=c (or A=xy) with respect to t; solve for the needed rate.

1. Principle. If two quantities are locked by a geometry or product rule, their speeds are locked too — differentiate the lock, don’t invent a new theory.
2. Model. e.g. A=x·y with A′ known → x′y + xy′ = A′; or ladder x²+y²=L² → xx′+yy′=0.
3. Prediction / Answer. Square oil slick A=πr², A′=2 m²/s at r=3 → 2πr r′=2 → r′=1/(3π) m/s. Transfer of chain + product into a moving job.

V2 — Light DE job: y′=ky → y=y₀e^{kt}.

Builds on: exp-log Tool (e^{kx})′=k e^{kx} and “rate ∝ amount.” Span +1: same DE reading, now a prediction job with data.

Tool gained: fit k from two measurements (or half-life / doubling), then predict y(t) — still the light DE only.

1. Principle. When growth speed tracks current size, the solution family is exponential — you already earned why (e^{kt})′ = k e^{kt}.
2. Model. y′=ky ⇒ y=y₀ e^{kt}. Find k from y(t₁)/y₀ = e^{kt₁}.
3. Prediction / Answer. Culture doubles in 5 h → k=ln2/5. Predict amount at 15 h: 8× start. Full DE methods (integrating factor, systems) stay out of span.

V3 — Mixed: optimize a cost that also grows.

Builds on: V1 related-rate / apps extrema + V2 light DE. Span +1: one job that needs both Tool families — still not full DE theory.

Tool gained: Kap 2 transfer pack — extrema check and light exponential prediction on the same board.

1. Principle. Real boards mix “best size now” (f′=0 + ends) with “how fast the stock grows” (y′=ky). Wrap means you can switch without leaping into DE courses.
2. Model. Choose x to minimize C(x) via C′; separately forecast stock S(t)=S₀e^{kt}; report both for the same shift plan.
3. Prediction / Answer. Fence-optimal pen width from A′=0, plus overnight bacterial count from doubling time — two Kap 2 Tools, one briefing.