Tangent Line: Touches a curve at exactly one point.
Secant Line: Touches a curve at two or more points.
Challenges
Finding the slope of a line at a single tangent point is challenging because it typically requires two points.
Approach to Solve Tangent Line Problems
Use a secant line to approximate the tangent line by getting closer to the point.
Choose two points on the curve that are very close to the point of interest.
As the points get closer, the approximation of the slope becomes more accurate.
Example Problem
Task: Find the equation of a tangent line.
Given: Parabola ( y = x^2 ) with a point at ( P(1, 1) ).
Steps:
Use the point ( P(1, 1) ) and approach it with a point ( Q(x, y) ).
Use the formula for slope ( \frac{y_2 - y_1}{x_2 - x_1} ) and substitute ( y = x^2 ).
Approaching x from both sides (left-hand side and right-hand side) to find the limit.
Calculations
Left Hand Approach: Choose values like 0.9, 0.99, 0.999 approaching 1.
Right Hand Approach: Choose values like 1.1, 1.01, 1.001 approaching 1.
Both approaches should converge to the same slope value, confirming the actual slope ( M ).
Finding the Equation of a Line
Once slope ( M ) is found, use the point-slope form to find the equation ( y = mx + b ).
Plug in the known point ( (1,1) ) to find ( b ).
Final equation: ( y = 2x - 1 ).
Velocity and Limits
Velocity Problem
Similar technique applies for velocity.
Use initial and final positions to find average velocity.
Formula: ( s(t) = 4.9t^2 ).
Find velocity at a specific time, e.g., ( t = 5s ).
Average Velocity
Calculate over intervals, e.g., ( t = 5 ) to ( t = 5.1 ) seconds.
Use ( \frac{s(5.1) - s(5)}{0.1} ) to find approximate velocity.
Conclusion
The process of finding a tangent line or velocity involves approaching from both sides to ensure the calculated limit is a true representation of the slope or velocity.
Practice these techniques using different functions and intervals to solidify understanding.