MHB Advanced Functions Average vs. Instantaneous velocity

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The discussion focuses on the concept of average velocity over short time intervals, specifically examining intervals [2, 2.01] and [1.99, 2]. It highlights how the average velocity can be calculated by dividing the distance moved by the time taken, illustrating this with a specific example of moving from position 2 to 2.01 over 0.01 hours. The conversation suggests a relationship between average velocity on an interval and instantaneous velocity at a midpoint, prompting deeper reflection on the meaning of instantaneous velocity. It raises questions about the challenges in defining velocity for a specific interval, emphasizing the importance of understanding average values. Overall, the discussion encourages active engagement with the mathematical concepts involved.
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What do the average velocities on the very short time intervals [2,2.01] and [1.99,2] approximate? What relationship does this suggest exist between a velocity on an interval [a,b] and a velocity near t=a+b/2 for this type of polynomial?
 
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That looks very much like a question that is asking you to think about what happens when you calculate an "average value" in order to lead you think about what the phrase "instantaneous velocity" could mean! You need to do it, not have someone else do it for you!
If at time t= 1.99 you are at point 2 and at time 2 you are position 2.01, you have moved from 2 to 2.01 so have moved a distance 2.01- 2= 0.01. And you did that in time interval 2- 1.99= 0.01.

If you move a distance 0.01 (km, say) in 0.01 (hours, say) what was your average velocity in that time interval?

Since velocity, in this way, is "the distance moved in a given time interval divided by the length of that time interval", do you see the problem with even defining "velocity at a given interval"?
 
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