Applications of Derivatives

Derivatives solve real design problems. Learn increasing/decreasing tests, stationary points, local and global extrema, concavity and inflection points, curve sketching, optimisation, related rates, Newton's method, linearisation, Taylor approximations and sensitivity analysis.

Where is power maximum? What dimensions minimise material? How fast does a water level rise? How do we solve an equation numerically? All these use derivatives.

An analogy

At the top of a hill (maximum) or the bottom of a valley (minimum) the ground is momentarily flat — the derivative is zero there.

Shape of a graph

f′ > 0: increasing; f′ < 0: decreasing; f″ > 0: concave up; f″ < 0: concave down; inflection where concavity changes.

Optimisation procedure

Define the variable and objective, express constraints, reduce to one variable, differentiate, set to zero, test (second derivative or endpoints), and check physical feasibility.

Related rates

Differentiate a relation between quantities with respect to time, then substitute known rates: ladders sliding, tanks filling, radar tracking.

Approximations and Newton's method

Linear and Taylor approximations underpin small-signal models of transistors and linearised control. Newton's method converges quadratically near a simple root but can diverge with poor starting points or f′ ≈ 0.

Formulas

• Second-derivative test: f′(c) = 0: f″(c) > 0 ⇒ minimum; f″(c) < 0 ⇒ maximum
• Linearisation: f(x) ≈ f(a) + f′(a)(x − a)
• Taylor polynomial: f(x) ≈ Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!
• Newton's method: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ)
• Relative sensitivity: S = (∂y/∂p)(p/y)

Worked example: Open-box optimisation

Given: Square sheet 30 cm; cut squares x from corners

Find: x for maximum volume.

• V = x(30 − 2x)²
• V′ = (30 − 2x)(30 − 6x) = 0 → x = 5 (x = 15 gives zero volume)
• V″(5) < 0 → maximum

Answer: x = 5 cm, V = 2000 cm³

Worked example: Newton's method for √2

Given: f(x) = x² − 2, x₀ = 1

Find: Three iterations.

• x₁ = 1.5
• x₂ = 1.416667
• x₃ = 1.414216

Answer: ≈ 1.414216 (6 correct digits after 3 steps)

Common mistakes

• Wrong: Ignoring endpoints in optimisation. — Right: Global extrema may be at boundaries.
• Wrong: Assuming f′ = 0 is always an extremum. — Right: x³ at 0 is an inflection point.

Applications

• Maximum power transfer: P = V²R/(R + R_s)² is maximum at R = R_s.
• Minimum material: Can and tank dimensions.
• Small-signal models: Linearising non-linear devices.

Key takeaways

• f′ = 0 → candidates; test with f″ or endpoints.
• Related rates via implicit time derivatives.
• Linear/Taylor approximations.
• Newton: xₙ₊₁ = xₙ − f/f′.

Going further

• Multivariable optimisation: Gradient = 0 and Hessian tests (M11).
• Taylor remainder: Error bounds for approximations.
• Sensitivity analysis: Robust design and tolerancing.