Applications of Derivatives
Applications of Derivatives
The exam tests rates of change, equations of tangents and normals, intervals of increase and decrease, and local or absolute maxima and minima, including simple word problems.
Core rules
- Rate of change: if y = f(x), then dy/dt = f′(x)·dx/dt; relate the given rates carefully.
- Tangent at (x₁, y₁): slope m = f′(x₁), equation y − y₁ = m(x − x₁). Normal slope = −1/m when m ≠ 0.
- f is increasing where f′(x) > 0 and decreasing where f′(x) < 0; solve f′(x) = 0 for stationary points and test sign changes.
- Second derivative test: f′(c)=0 and f″(c)<0 gives a local maximum; f″(c)>0 gives a local minimum.
- For absolute extrema on [a,b], evaluate f at all critical points and at both endpoints.
Worked example
Find the intervals of increase and decrease of f(x) = x³ − 3x² + 1.
f′(x) = 3x² − 6x = 3x(x − 2). The critical points are x = 0 and x = 2. Testing signs: f′ > 0 on (−∞, 0), f′ < 0 on (0, 2), and f′ > 0 on (2, ∞). So f increases on (−∞, 0) ∪ (2, ∞) and decreases on (0, 2). At x = 0, f = 1 is a local maximum; at x = 2, f = −3 is a local minimum.
Traps
- Giving closed intervals for monotonicity when the derivative vanishes at the endpoints.
- Stopping at f′(x)=0 without checking the sign change.
- Confusing local with absolute extrema; closed-interval problems require endpoint values.
- Using the normal slope as f′(x₁) instead of −1/f′(x₁).
- Ignoring units and the chain rule in related-rates problems.