Chapter 4: Applications of Differentiation

Students learn how to locate local maxima and minima via the First and Second Derivative Tests, and how these concepts translate to practical optimization problems in physics, engineering, and economics. The chapter covers modeling motion with derivatives, connecting position, velocity, and acceleration to create detailed motion profiles. It also introduces related rates problems, where multiple quantities change over time, and shows how implicit differentiation is used to relate their rates of change. Real-world examples include expanding circles, rising ladders, and inflating balloons, demonstrating calculus in action. The section on linear approximations and differentials explains how tangent lines can be used to estimate function values and analyze error propagation. The Mean Value Theorem is presented with both its geometric interpretation and its implications for function behavior. The chapter also addresses L’Hôpital’s Rule for evaluating limits of indeterminate forms such as 0/0 and ∞/∞, extending the techniques introduced earlier. Finally, the chapter emphasizes how these derivative-based tools interconnect—optimization relies on extrema analysis, related rates require implicit differentiation, and L’Hôpital’s Rule draws on limit concepts—creating a unified problem-solving framework.