Comprehensive notes on uses of derivatives and optimization

Introduction

  • Optimization is central to economic models and social science models; used to derive supply/demand, profit maximization, and cost minimization.
  • Tools of optimization include constructing graphs of functions, understanding how derivatives reveal slopes and curvature, and identifying critical points.
  • In these notes, we start with graphing arbitrary one-variable functions via first- and second-order derivatives, then move to optimization: locating maxima/minima, and the role of concavity/convexity and constraints.

Graphing functions of one variable using derivatives

  • Goal: graph arbitrary f(x) using information from f'(x) and f''(x).
  • Rule 2.1 (Increasing/Decreasing via first derivative):
    • If f'(x_0) > 0, then f is increasing at x0 (there exists an interval around x0 where f(x1) < f(x2) for x1 < x2).
    • If f′(x0)<0f'(x_0) < 0, then f is decreasing at x0 (there exists an interval around x0 where f(x1) > f(x2) for x1 < x2).
  • Rule 2.1 is silent when f′(x0)=0f'(x_0)=0; such points are called critical points.
  • Definition 2.1: Critical point
    • If f′(x0)=0f'(x_0)=0, then x0 is a critical point of f.
  • Definitions 2.2/2.3: Local maxima/minima
    • Local maximum: there exists I=(a,b) with a<x0<b such that $$f(x) \