Average Rate of Change

🎯 What is Average Rate of Change?

The average rate of change measures how much a function's output changes per unit of input over a specific interval. It's essentially the slope of the secant line connecting two points on a function's graph.

This concept is fundamental in:

• 📈 Analyzing function behavior
• 🚗 Calculating velocity and acceleration
• 💰 Economics and business analysis
• 🔬 Scientific data interpretation

📋 Step-by-Step Process

1. Identify the interval [a, b] you want to analyze
2. Calculate f(a) - the function value at the left endpoint
3. Calculate f(b) - the function value at the right endpoint
4. Find the difference: f(b) - f(a)
5. Find the interval length: b - a
6. Divide the difference by the interval length

🔍 Key Concepts

Geometric Interpretation

The average rate of change represents the slope of the secant line between two points on a curve. This connects the algebraic concept to visual geometry.

Relationship to Slope

For linear functions, the average rate of change is constant and equals the slope. For non-linear functions, it varies depending on the chosen interval.

Units and Interpretation

The units of average rate of change are always output units per input unit. For example, miles per hour, dollars per year, or degrees per minute.

🌟 Real-World Applications