Applications: Related Rates, Extrema & Curve Sketching

Related rates

Two changing quantities linked by geometry or a formula. Differentiate both sides with respect to time t, plug in known rates, solve for the unknown rate.

• Draw and label; write the linking equation.
• Differentiate implicitly: d/dt of both sides.
• Substitute known values only after differentiating.

Extrema and the first derivative test

Critical points: f′(x) = 0 or f′ undefined (in domain). Test sign changes of f′ or use the second derivative test when f″ exists.

Optimisation checklist

• Express the quantity to optimise as a function of one variable.
• Identify the closed interval or realistic domain.
• Evaluate at critical points and endpoints.
• Interpret the answer with units and context.