Applications of Derivatives — Curve Sketching & Critical Points

The Derivative as a Tool for Analysis

The first and second derivatives together give a complete picture of a function's behaviour: where it rises and falls, where its extrema are, and how it curves. This analysis — sometimes called curve sketching — is one of the most powerful applications of differential calculus.

Increasing and Decreasing Functions

• f'(x) > 0 on an interval → f is increasing there.
• f'(x) < 0 on an interval → f is decreasing there.
• f'(x) = 0 → critical point (potential max or min).

Process: find f'(x), solve f'(x) = 0 for critical points, test the sign of f' in each interval between critical points.

Finding Local Extrema

First Derivative Test: if f' changes from + to − at c, there is a local maximum at c. If f' changes from − to + at c, there is a local minimum. No sign change → neither.

Second Derivative Test: if f'(c) = 0 and f''(c) > 0 → minimum; f''(c) < 0 → maximum.

Mean Value Theorem

The MVT states: if f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) where f'(c) = (f(b)−f(a))/(b−a). In plain terms, the instantaneous rate equals the average rate at some point. The MVT is the theoretical underpinning of every "increasing/decreasing on an interval" argument — it is what allows us to conclude that positive derivative implies increasing function. Without MVT, that intuition cannot be proved rigorously.

The MVT has a practical implication: if you drive 100 miles in 1 hour, your speedometer must have read exactly 100 mph at some point during the journey, even if your speed varied constantly. Speed cameras on highways exploit a discrete version of this — average speed between two checkpoints determines whether you exceeded the limit at some point.

Linear Approximation

The derivative at a point defines the tangent line, which provides the best linear approximation to the function near that point. f(x) ≈ f(a) + f'(a)(x−a). This is called the linearisation of f at a. Engineers use linear approximations constantly: small-angle approximation sin θ ≈ θ (from d/dx[sin x] = cos(0) = 1), resistance calculations for small perturbations, and control system design all rely on linearisation around an operating point.