Applications of Derivatives — Max/Min, Related Rates, MVT

GRE Math Subject · Axiom Academy

LESSON Applications of Derivatives — Max/Min, Related Rates, MVT Real-world problem solving with derivatives Step 1: Optimization (Max/Min) 1. Set up the objective function f(x) to maximize/minimize 2. Find f'(x) and set it equal to zero: f'(x) = 0 4. Use second derivative test: f''(x) > 0 ⟹ local min, f''(x) < 0 ⟹ local max 5. Check boundary points and compare values Second Derivative Test: If f'(c) = 0 and f''(c) exists: f''(c) = 0 ⟹ inconclusive (test fails) GRE Example: Maximize area A = xy subject to constraint x + 2y = 100. Solve x = 100 - 2y, so A(y) = (100 - 2y)y = 100y - 2y². Then dA/dy = 100 - 4y = 0 ⟹ y = 25, x = 50. Max area = 1250. Related rates problems involve quantities that change with time. The key: relate the rates using implicit differentiation. 1. Draw a diagram and identify all variables 2. Write an equation relating the variables (often using geometry) 3. Differentiate both sides with respect to time t 4. Substitute known values and solve for the unknown rate Classic Example: A ladder 10 ft long leans against a wall. The base slides away at 2 ft/s. How fast is the top sliding down when the base is 6 ft from the wall? Setup: x² + y² = 100 (where x = distance base to wall, y = height on wall) Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0 When x = 6: 36 + y² = 100 ⟹ y = 8 Given dx/dt = 2: 2(6)(2) + 2(8)(dy/dt) = 0 ⟹ dy/dt = -1.5 ft/s Mean Value Theorem (MVT): If f is continuous on [a,b] and differentiable on (a,b), then there exists c ∈ (a,b) where:

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