Applications of derivatives - Mathematics: Analysis & Approaches IB Study Notes

Applications of derivatives form a cornerstone of IB Mathematics: Analysis & Approaches, transforming abstract calculus into powerful problem-solving tools.

Key Definitions:

Tangent and Normal Lines: The derivative f'(a) represents the gradient of the tangent line at point (a, f(a)). The tangent equation is: y - f(a) = f'(a)(x - a). The normal line is perpendicular to the tangent, with gradient -1/f'(a), giving equation: y - f(a) = -1/f'(a) × (x - a).

Increasing and Decreasing Functions: A function is increasing on an interval when f'(x) > 0 (gradient positive), and decreasing when f'(x) < 0 (gradient negative). When f'(x) = 0, the function has stationary points.

Stationary Points: Points where f'(x) = 0. These include:

• Local maximum: f'(x) changes from positive to negative
• Local minimum: f'(x) changes from negative to positive
• Point of inflection: f'(x) doesn't change sign (gradient zero but continues in same direction)

Second Derivative Test: For stationary point at x = a:

• If f''(a) > 0: local minimum (concave up, ∪ shape)
• If f''(a) < 0: local maximum (concave down, ∩ shape)
• If f''(a) = 0: test is inconclusive; use first derivative test

Optimization: Finding maximum or minimum values of functions subject to constraints. Essential formula: dA/dx = 0 identifies critical points.

Rates of Change: The derivative dy/dx represents instantaneous rate of change. For related rates: use chain rule to connect multiple changing quantities.

Memory Aid (TINS): Tangents, Increasing/decreasing, Normals, Stationary points—the four fundamental applications!

Understanding derivative applications becomes intuitive through real-world contexts.

Optimization in Business: Imagine a company manufacturing smartphones. The profit function P(x) = -2x² + 800x - 5000 (where x = units in thousands) models revenue minus costs. Finding dP/dx = 0 reveals the production level maximizing profit. At x = 200 (200,000 units), profit peaks—produce fewer, you're leaving money on the table; produce more, costs exceed revenue gains. This is exactly how businesses use calculus for decision-making.

Motion Analysis: Consider a rocket's height h(t) = -5t² + 100t + 20 meters at time t seconds. The derivative h'(t) = -10t + 100 gives velocity. When h'(t) = 0 (at t = 10s), the rocket reaches maximum height (520m) before falling. The second derivative h''(t) = -10 (constant negative acceleration due to gravity) confirms this is a maximum. Athletes, engineers, and space agencies use these principles daily.

Related Rates in Nature: Picture a spherical balloon inflating. Volume V = (4/3)πr³ and radius r both change with time. When air flows in at dV/dt = 100 cm³/s, how fast does radius grow? Using chain rule: dV/dt = dV/dr × dr/dt = 4πr² × dr/dt. At r = 5cm, solving gives dr/dt = 100/(4π×25) ≈ 0.318 cm/s. This models everything from tumor growth to bubble dynamics.

Tangent Applications: GPS navigation uses tangent lines to approximate curved roads as straight segments. When your phone calculates "3.2 km ahead," it's using linear approximation: f(x) ≈ f(a) + f'(a)(x-a), treating the road as its tangent line momentarily.

Analogy: Think of derivatives as a "mathematical microscope"—zooming into curves until they look straight, revealing their instantaneous behavior.