Application of Derivatives
Application of Derivatives turns f′ and f″ into geometry and optimisation: rates of change, monotonicity, tangents and normals, small approximations, and maxima–minima on an interval. Word problems on maximising area, volume or profit are standard, so reduce the expression to one variable before differentiating.
Core tools
- Rate of change: if y = f(x) and x = g(t), then dy/dt = f′(x)·dx/dt.
- f is strictly increasing where f′(x) > 0 and strictly decreasing where f′(x) < 0, on an interval.
- Tangent slope at (x₁, y₁) is m = dy/dx; normal slope is −1/m. Tangent: y − y₁ = m(x − x₁).
- Critical points satisfy f′(x) = 0 or f′ undefined. Second derivative test: f″(c) < 0 gives a local maximum, f″(c) > 0 gives a local minimum, f″(c) = 0 is inconclusive.
- Absolute extrema on [a, b]: evaluate f at every critical point inside and at both endpoints.
Worked example
A wire of length 36 cm is bent into a rectangle. Find the maximum area. Let one side be x; the other is 18 − x. Then A(x) = x(18 − x) = 18x − x². A′(x) = 18 − 2x = 0 gives x = 9, and A″(x) = −2 < 0, so x = 9 is a maximum. The rectangle is a 9 cm × 9 cm square with area 81 cm². The endpoint values A(0) = A(18) = 0 confirm this is the absolute maximum.
Traps
- Reporting only local extrema when the question asks for the absolute maximum or minimum.
- Concluding a maximum from f″(c) = 0.
- Forgetting the negative reciprocal when writing the normal.
- Differentiating a two-variable constraint instead of eliminating one variable first.