Academia Aeternum
तमसो मा ज्योतिर्गमय
Wisdom · Knowledge · Eternity
Rates, Monotonicity & Optimisation — All Application of Derivatives Solutions
4 exercise files · 90 total questions
\(f ext{ increasing on }I \iff f'(x) \geq 0\ orall x\in I\)\( ext{Critical point: } f'(c)=0 ext{ or }f'(c) ext{ undefined}\)\( ext{2nd derivative test: } f''(c)>0\Rightarrow ext{local min};\ f''(c)<0\Rightarrow ext{local max}\)\(\text{Related rates: differentiate the connecting equation w.r.t. }t\)Step 1 — Related rates: write the equation linking the quantities FIRST, then differentiate both sides w.r.t. time. Step 2 — Monotonicity: find f'(x), solve f'(x)=0 for critical points, test the sign of f'(x) in each sub-interval. Step 3 — Maxima/minima: locate critical points, apply the second derivative test (or first-derivative sign change) to classify each. Step 4 — Optimisation word problems: express the quantity to optimise as a single-variable function using the given constraint, then apply Step 3.
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