Lecture Notes: Understanding Slopes and Tangent Lines

Key Concepts

Slope (M): The slope is represented by M in equations.
Tangent Line: A line that touches a curve at exactly one point. The slope of the tangent line at the point of contact is equal to the slope of the curve at that point.
Secant Line: A line that intersects a curve at two or more points. Used to approximate the slope of a curve when only one point is known.
Rise over Run Formula: Used to calculate the slope of a line between two points: ( \frac{y_2 - y_1}{x_2 - x_1} ).

Problem: To find the slope of a tangent line touching a curve at one point when the usual slope formula requires two points.
Approach: Use a secant line to approximate, move closer to the point, and eventually find the true slope by getting infinitely close.
Example: Using the function ( y = x^2 ) to find the slope at point P (1, 1).
- Method: Choose a nearby point Q and use ( \frac{y_2 - y_1}{x_2 - x_1} ) with each iteration getting closer to the point P.

Limits: Used to find the slope as a point approaches another point on the curve.
- Left-Hand Approach: Approach the point from lower values, e.g., 0.9, 0.99.
- Right-Hand Approach: Approach from higher values, e.g., 1.1, 1.01.
- Conclusion: If both approaches yield the same slope value, the limit exists.

Equation: ( y = 2x + b )
- Steps to Find b: Use the point P (1, 1) to solve for b in the equation after finding the slope.
- Result: Equation of the tangent line is ( y = 2x - 1 ).

Scenario: Ball dropped from 450m, find velocity after 5 seconds.
Given Formula: Distance ( s(t) = 4.9t^2 ).
Approach: Similar to finding tangents, use intervals to approximate velocity change over time.
- Technique: Use average velocity ( \frac{S(t_2) - S(t_1)}{t_2 - t_1} ) and reduce the time interval to improve accuracy.

Importance of Practice: Emphasized need for frequent practice with calculators and understanding different types of functions.
Homework: Tackle various problems using tangent and velocity concepts to solidify understanding.