Kenya Leonard

Answered

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?

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?

Answer & Explanation

Kendrick Jacobs

Expert

2022-07-17Added 16 answers

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).

$\frac{\mathrm{\Delta}y}{\mathrm{\Delta}x}$

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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