Overview
This lecture introduces the main ideas of Calculus 1, focusing on slopes, rates of change, and the foundational concept of the derivative, with an emphasis on calculating and understanding average rate of change.
Introduction to Calculus and Rates of Change
- Calculus 1 centers on understanding slopes and how things change over time (rates of change).
- Key course goal: Learn how to describe and compute how fast something changes at a specific instant, called the derivative.
- The instantaneous rate of change (how fast something is changing at a specific moment) is more complex than the average rate of change.
Average Rate of Change (AROC)
- Average rate of change finds how much a quantity changes over a given interval.
- Formula: AROC from x = a to x = b is [f(b) - f(a)] / (b - a), using function notation.
- "f(b)" is the y-value (output) when x = b, and "f(a)" is the y-value when x = a.
- Numerator, f(b) - f(a), represents the change in y (vertical distance, Δy, or "rise").
- Denominator, b - a, represents the change in x (horizontal distance, Δx, or "run").
- The average rate of change is the slope of the line between two points (a, f(a)) and (b, f(b)).
Visualizing Average Rate of Change
- On a graph, the average rate of change is the slope of the secant line connecting two points on a curve.
- For curved graphs, AROC estimates how steep the curve is between two points.
Pre-Calculus Review & Algebra Skills
- Function notation is essential: f(x) represents the output (y-value) for input x.
- Strong algebra skills are vital for success in calculus—expect lots of algebraic manipulation.
Key Terms & Definitions
- Rate of Change — How fast a quantity changes with respect to another (e.g., distance over time).
- Average Rate of Change (AROC) — The total change in a function's value divided by the change in input over an interval.
- Instantaneous Rate of Change — The rate at which a function changes at a single point (the derivative).
- Derivative — The mathematical concept for instantaneous rate of change.
- Function Notation — Writing a function as f(x), where x is input and f(x) is output.
- Secant Line — A line connecting two points on a curve, representing average rate of change between them.
Action Items / Next Steps
- Review function notation and algebra skills from pre-calculus.
- Practice calculating average rate of change from formulas, graphs, and tables.
- Prepare for upcoming examples and problems involving rates of change.