Approximating instantaneous rate of change word problem  Differential Calculus  Khan Academy  Summary and Q&A
TL;DR
This video discusses a motorcyclist's position and velocity over time, including calculations for average velocity and approximation of instantaneous velocity.
Questions & Answers
Q: How is average velocity calculated for a given time interval?
Average velocity is determined by dividing the change in distance by the change in time. In the video, the example interval between 1.5 and 2 seconds had a distance change of 11.5 meters and a time change of 0.5 seconds, resulting in an average velocity of 23 meters per second.
Q: How can the instantaneous velocity be approximated?
To approximate instantaneous velocity, one can average the slopes of tangent lines between nearby intervals. In the video, the slopes at intervals 1.52 seconds and 22.5 seconds were averaged to obtain an approximation of 27.4 meters per second.
Q: What is the equation of a tangent line in pointslope form?
The equation of a tangent line can be written in pointslope form, where the slope is multiplied by the difference in the independent variable (t) and added to the ycoordinate (S) of a point on the line. In the video, the equation was written as S  30.2 = 27.4(t  2).
Q: What does the table in the video depict?
The table displays the position of a motorcyclist at different time intervals. It shows the distance traveled (S) in meters and the corresponding time (t) in seconds. This information is used to calculate velocities and approximations of instantaneous velocity.
Summary & Key Takeaways

The table shows the position of a motorcyclist at various time intervals, with distance traveled measured in meters and time measured in seconds.

The video explains how to calculate average velocity using the change in distance and change in time for a given interval.

The video also demonstrates how to approximate the instantaneous velocity at a specific time by averaging the slopes of tangent lines between nearby intervals.

Finally, it provides a stepbystep example of writing the equation of a tangent line in pointslope form using the calculated slope and a point on the line.