Overview
This lecture covers the concept of rates of change, focusing on key terms like slope, how to calculate average rate of change from graphs, tables, and functions, and how rates of change relate to increasing or decreasing behavior in functions.
Rate of Change Concepts
- Rate of change measures how a quantity changes with respect to another (often time).
- Slope is the rate of change between two points on a line.
- Average rate of change is calculated over an interval, not at a single point.
- In functions, rate of change describes how the dependent variable (Y) changes as the independent variable (X) changes.
Calculating Average Rate of Change
- The formula: Average Rate of Change = (f(b) - f(a)) / (b - a).
- From a table: Use corresponding function values at two X-values and divide their difference by the difference in X.
- From a graph: Read Y-values at given X-points, apply the formula, and interpret as the slope of the connecting line.
- With function notation: Substitute the X-values into the function, then apply the formula.
Examples
- Example with position: If a car travels 60 miles in one hour, average rate of change = 60 miles/hour.
- For f(x) from a graph: If f(2) = 3, f(-1) = 0, rate of change from -1 to 2 is (3-0)/(2-(-1)) = 1.
- Table Example: From x=4 to x=8, if f(8)=16 and f(4)=6, rate of change = (16-6)/(8-4) = 2.5.
- Example with variable: (1+t)² – 1² over t yields (2t + t²)/t = 2 + t.
Increasing and Decreasing Functions
- A function is increasing where the rate of change is positive (graph moves upward left to right).
- A function is decreasing where the rate of change is negative (graph moves downward left to right).
- Use open intervals to describe regions where a function is increasing or decreasing.
- At endpoints (where change switches), the function is neither increasing nor decreasing.
Key Terms & Definitions
- Rate of Change — How one variable changes in relation to another.
- Slope — The rate of change between two points, calculated as rise over run.
- Average Rate of Change — (f(b) - f(a)) / (b - a); the change in Y over the change in X for an interval.
- Increasing Function — A function whose values rise as X increases.
- Decreasing Function — A function whose values fall as X increases.
Action Items / Next Steps
- Complete questions 1–5 on the Section 1.3 worksheet.
- Finish Section 1.3 and check answers with the provided key.