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By definition a rate is a ratio of two different units. let us say those units refers to change in x and y of a function. But instant means no change and if there is no change there is nothing to ratio for. Surely instantaneous rate of change is senseless literally. But why do they keep stating it as instantaneous rate of change (Rate of change at a moment) instead of best linear approximation at a point?. Should it be fitting to include it in physics and not in math?

Edz
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  • It's the limit value as you calculate rate of change over shorter and shorter intervals around the point of interest. – Joffan Jul 04 '18 at 14:36
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    Instantaneous rate of change is not a ratio (between two finite quantities) but the limit of a ratio. See instantaneous velocity. – Mauro ALLEGRANZA Jul 04 '18 at 14:37
  • Suppose you treat speed (or velocity if you prefer) as a rate of change in distance compared with time. If you accelerate so your speed keeps increasing, is it meaningful to ask at what point in time your speed was a particular value? Would that be an instantaneous rate of change? – Henry Jul 04 '18 at 14:40
  • Possible duplicate https://math.stackexchange.com/questions/2136779/calculus-why-do-we-define-rate-of-change-as-dy-dx/2136831#2136831 – Ethan Bolker Jul 04 '18 at 14:45

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It just means the 'rate' of something in an instant in time. It is usually found as the limit of some function that takes the difference between to 'rates' or points or whatever over decreasing increments of time. It can often be found by taking the first of second differential of some function. For example the derivative (differential) of a distance function gives the rate of speed and the second derivative is the rate of acceleration.

poetasis
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  • But how can you have rate of change at an instant if there is no change at an instant? – Edz Jul 04 '18 at 14:43
  • You start with an equation that does measure things over time and then figure out how to find the LIMIT of that equation (function) over shorter and shorter time intervals. Eventually, the interval becomes zero by some mathematical manipulation and it may not be intuitive but if it is mathematically consistent, it is true and reflects something you can measure or 'predict' in the real world. – poetasis Jul 04 '18 at 14:47
  • Becomes zero?No.it gets closer to zero. But that is not instant. Yes a difference quotient measures ratio of change in x and y but so far as there is change. – Edz Jul 04 '18 at 14:52
  • I took liberties for description. I just meant that we end up with a function that predicts some value for some time 't' and meant to suggest that it would be 'as if' the time interval measured were zero. – poetasis Jul 04 '18 at 14:55
  • So what you mean to say is. If difference quotient approach some value "Limit" while simultaneously approaching a geometrical property of "Nothing" which means no interval no change a point Since they both approach something simultaneously (a property of constant slope and a geometrical property of a point.)That " Nothing"(Instant) equals "Limit" ? – Edz Jul 04 '18 at 15:07
  • You need to look up limit in a precalc or calculus book or in wikipedia or someplace. It's too much for an extended discussion here. Try this to start: https://en.wikipedia.org/wiki/Limit_of_a_function – poetasis Jul 04 '18 at 15:34