Overview
This lecture covers the concept of rates of change, focusing on average rate of change, slope, calculations using graphs, tables, and functions, and how these relate to intervals where functions increase or decrease.
Rate of Change & Slope
- Rate of change measures how one variable changes in relation to another.
- Slope is the rate at which the dependent variable (y) changes per one unit of the independent variable (x).
- Average rate of change is calculated as the change in function values divided by the change in input values.
Calculating Average Rate of Change
- Formula: Average Rate of Change = [f(b) - f(a)] / (b - a).
- The slope between two points on a graph represents the average rate of change.
- From tables: use values from two input points to calculate the difference quotient.
- From a function: substitute the inputs into the function, find their outputs, and apply the formula.
- With variables, expand and simplify expressions as needed before dividing.
Interpreting Increasing and Decreasing Intervals
- A function is increasing on intervals where the average rate of change is positive.
- A function is decreasing on intervals where the average rate of change is negative.
- Use open intervals to specify where a function is increasing or decreasing.
- At points where transitions occur (peaks/valleys), the function is neither increasing nor decreasing momentarily.
Applying Concepts with Tables and Graphs
- Examine function values between points: if values rise, the function is increasing; if they fall, it is decreasing.
- Mark intervals of consistent increase/decrease based on table or graph data.
Key Terms & Definitions
- Rate of Change — How much one variable changes per unit change in another variable.
- Slope — The measure of steepness, calculated as “rise over run.”
- Average Rate of Change — The total change in the dependent variable divided by the total change in the independent variable over an interval.
- Increasing Interval — Interval where function values rise as the input increases.
- Decreasing Interval — Interval where function values fall as the input increases.
Action Items / Next Steps
- Complete Section 1.3 worksheet questions 1–5.
- Review and check your answers using the answer key.