- #1

DumpmeAdrenaline

- 78

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Find the average speed between t=1 and t=1+h where h is any real number except 0.

Distance traveled/Time it takes to travel the distance =16(1+h)^2-16=16+32h+16h^2-16/h=32+16h

We obtained the correct answer for the wrong problem. I have never fully understood what the instantaneous rate of change of any quantity meant. What is an instant? From experience, we say that an instant is an extremely tiny interval, but tiny to whom? It's quite subjective. Also, in keeping the definition of average rate of change to understand the instantaneous rate of change we require the knowledge at two instants, How can we recourse to a definition that requires the measurement at two instants to understand what happens at one?

Accepting that an instant is an extremely time interval and that nothing drastic happens over that tiny interval we observe that the average rate of change serves as a better approximation as the time interval shrinks but that we loose all information if no time transpires (we wind up with 0/0).

32+16h ≅ 32 we sense that 16h approaches as h approaches 0 and 32+16h tends to 32.

Why do we say the limiting value is equal to 32 and not approximately 32?

It seems to me that at first we use h≠0 to steer away from 0/0 and then we obtain the limiting value by plugging in the only value we are not allowed to plug.