Kenya Leonard

2022-07-16

Rate of change vs. average rate of change of a function
What is the difference between the both? I am trying to find a separate definitions for both in a textbook, but it only defines average rate of change.
I know that average rate of change is:
$\frac{change\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}y}{change\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}x}$
But what is just rate of change then?

Kendrick Jacobs

Expert

The average rate of change is defined over some finite interval $\mathrm{\Delta }x$ to be
$\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$
The rate of change is the rate at which the function changes at one particular point and is found by taking the limit
$\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$
Note that in the case of a linear function, y=mx+c, the rate of change and the average rate of change are identical (they both equal m).

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