🧮 The Average Rate of Change Formula
Basic Formula
Where [a, b] is the interval and f(x) is the function
📐 Alternative Forms of the Formula
1. Coordinate Point Formula
Used when you have two coordinate points: (x₁, y₁) and (x₂, y₂)
2. Delta Notation
Where Δy = change in y-values and Δx = change in x-values
🔍 Step-by-Step Application
Example 1: Using Function Notation
Problem: Find the average rate of change of f(x) = x² + 3x from x = 1 to x = 4
Example 2: Using Coordinate Points
Problem: Find the average rate of change between points (2, 5) and (6, 17)
🎯 Common Applications
1. Physics Applications
Velocity: Average velocity = (final position - initial position) / time interval
Acceleration: Average acceleration = (final velocity - initial velocity) / time interval
2. Economics Applications
Profit Rate: Average profit rate = (profit change) / (time period)
Growth Rate: Average growth rate = (final value - initial value) / time period
3. Biology Applications
Population Growth: Average growth rate = (population change) / time interval
Reaction Rate: Average reaction rate = (concentration change) / time interval
⚠️ Important Notes
- Order Matters: Always subtract in the same order: (final - initial) / (final - initial)
- Units: The result has units of (y-units) per (x-units)
- Interpretation: Positive values indicate increasing function, negative values indicate decreasing function
- Geometric Meaning: Represents the slope of the secant line connecting two points
🔗 Related Concepts
Instantaneous Rate of Change: The limit of average rate of change as the interval approaches zero (derivative)
Slope: For linear functions, the average rate of change equals the slope
Secant Line: The line connecting two points whose slope is the average rate of change
📚 Practice Problems
Problem 1: Find the average rate of change of f(x) = 2x³ - x + 1 from x = 0 to x = 2
Problem 2: A ball is thrown upward. Its height h(t) = -16t² + 64t + 5. Find the average rate of change from t = 1 to t = 3 seconds.
Problem 3: The temperature T(h) = 70 - 3.5h where h is altitude in thousands of feet. Find the average rate of temperature change from 2000 to 8000 feet.