Lesson 22.1

Tangent Lines and Rates of Change

Transitioning from algebra's secant lines to calculus's tangent lines.

Introduction

A secant line connects two points on a curve and gives the average rate of change. A tangent line touches the curve at exactly one point and gives the instantaneousrate of change. The tangent line is the limit of secant lines as the two points merge.

Prerequisite Connection

You understand slope, limits, and function behavior.

Today's Increment

We connect average and instantaneous rate of change through limits.

Why This Matters

This is the conceptual foundation for derivatives—the heart of differential calculus.

Key Concepts

Average Rate of Change (Secant Line)

Slope between two points (a, f(a)) and (b, f(b)).

Instantaneous Rate of Change (Tangent Line)

Slope at a single point—this IS the derivative!

Tangent Line Equation

Worked Examples

Example 1: Average Rate of Change (Basic)

Find the average rate of change of from x = 1 to x = 3.

Average rate = 4

Example 2: Instantaneous Rate (Intermediate)

Find instantaneous rate of change of at x = 2.

Instantaneous rate at x=2 is 4

Example 3: Tangent Line Equation (Advanced)

Find the equation of the tangent line to at x = 2.

Point: (2, 4). Slope: 4 (from above).

Common Pitfalls

Confusing average vs. instantaneous

Average uses two points. Instantaneous uses a limit at one point.

Forgetting to take the limit

The tangent slope is a LIMIT, not just the difference quotient.

Real-World Application

Speedometer Reading

Your car's speedometer shows instantaneous velocity—the limit of average velocity as the time interval approaches zero. This is exactly the derivative of position!