Overview

Derivatives don’t just compute rates — they help analyze functions, optimize systems, and describe motion. This topic covers how to use derivatives to understand behavior such as increasing/decreasing intervals, extrema, concavity, inflection points, and more.

Key Concepts and Structures

• Critical Points: Occur where f'(x) = 0 or f'(x) is undefined.
• Increasing/Decreasing: If f'(x) > 0, the function is increasing; if f'(x) < 0, it is decreasing.
• Local Maxima and Minima: Use the First Derivative Test:
  • If f' changes from positive to negative → local max
  • If f' changes from negative to positive → local min
• Concavity: Determined by the second derivative f''(x)
  • If f''(x) > 0, the graph is concave up
  • If f''(x) < 0, the graph is concave down
• Inflection Points: Points where concavity changes (where f''(x) = 0 and sign changes)
• Tangent Line: The derivative gives the slope of the tangent line at a point
• Linear Approximation: L(x) = f(a) + f'(a)(x - a)
• Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = \frac{f(b) - f(a)}{b - a} for some c in (a, b)

Quick Tip

Practice Problems

1. Find intervals where f(x) = x^3 - 3x is increasing or decreasing.
   Show Solution

   Step 1: Find f'(x) = 3x^2 - 3

   Step 2: Set f'(x) = 0 → 3x^2 - 3 = 0 → x = ±1

   Step 3: Test sign of f' in intervals:

   • On (−∞, −1): f'(x) > 0 → increasing
   • On (−1, 1): f'(x) < 0 → decreasing
   • On (1, ∞): f'(x) > 0 → increasing

   Final Answer: Increasing on (−∞, −1) ∪ (1, ∞); Decreasing on (−1, 1)

2. Find points of inflection for f(x) = x^4 - 4x^2
   Show Solution

   Step 1: Find f''(x) = 12x^2 - 8

   Step 2: Set f''(x) = 0 → 12x^2 - 8 = 0 → x = ±√(2/3)

   Step 3: Test concavity changes around those points

   Final Answer: Inflection points at x = ±√(2/3)