Overview
This lecture demonstrates how to find instantaneous velocity using a limiting process, highlighting its similarity to tangent line problems and introducing the need for limits in calculus.
Instantaneous Velocity Concept
- Instantaneous velocity is the rate of change of position at a specific moment in time.
- Calculating velocity at a single instant seems paradoxical, as velocity typically requires two points.
- The calculation uses a limiting process by examining average velocities over smaller and smaller intervals.
Problem Example: Ball Thrown Upward
- A ball is thrown straight up from a 64 ft building at 48 ft/sec.
- The height function is ( H(t) = -16t^2 + 48t + 64 ), where ( t ) is time in seconds.
- The goal: Find the instantaneous velocity at ( t = 2 ) seconds.
Calculating Average Velocities
- Calculate average velocity over intervals ([2, t]) with ( t ) slightly greater than 2.
- Average velocity formula: ( \frac{H(t) - H(2)}{t - 2} ).
- ( H(2) = 96 ), so the average velocity simplifies to ( \frac{-16t^2 + 48t + 32}{t - 2} ).
Numerical Results & Interpretation
- As ( t ) approaches 2 (e.g., 2.1, 2.01, 2.001), average velocities approach (-16) ft/sec.
- At ( t = 2 ), the instantaneous velocity is exactly (-16 ) ft/sec.
- At this moment, the ball is 96 ft high and is falling (velocity is negative).
Importance of Limits
- The limiting process defines the exact instantaneous velocity.
- The same mathematical steps apply as in tangent problems.
- Understanding limits is necessary for rigor in calculus and for future topics like derivatives.
Key Terms & Definitions
- Instantaneous Velocity — The exact rate of change of position at a specific instant in time.
- Height Function ( H(t) ) — Models the ball’s height above ground as a function of time.
- Average Velocity — The change in height divided by the change in time over an interval.
- Limit — The value that a function or sequence "approaches" as the input (or index) approaches some value.
Action Items / Next Steps
- Prepare to learn informal and formal definitions of limits in the next sections.
- Review previous notes on secant and tangent lines for comparison.
- Practice similar velocity problems using the limiting process.