📊 Chapter 3: Applications of Derivatives

AP Exam Weight: 25-35% | Multiple Choice: 10-14 questions | Free Response: Major focus in several questions

📚 Table of Contents

1. Mean Value Theorem
2. Extrema & Optimization
3. Related Rates
4. Curve Analysis
5. L’Hôpital’s Rule

1. Mean Value Theorem 📊

Understanding MVT

The Mean Value Theorem connects average and instantaneous rates of change. Think of it as:

• A car’s average speed vs. instantaneous speed
• The exact moment you’re going the average speed
• A mathematical guarantee about derivatives
• A tool for proving other theorems

Statement

If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) where: $f'(c) = \frac{f(b)-f(a)}{b-a}$

Intuitive Understanding

• Left side: instantaneous rate at some point
• Right side: average rate over interval
• Must match at least once
• Like hitting your average speed during a trip

Verification Process

1. Check continuity on [a,b]