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The instantaneous rate of change is supposed to be the rate of change at a particular moment. How can it be a rate of change at a point because the rate of change is the speed at which a variable changes over a specific period of time and their is only 1 point, not 2 points.
And secondly is the instantaneous rate of change at a point the same as the rate of change of any 2 points on the whole tangent line. Could someone explain what instantaneous rate of change is and what it represents?
Let $f$ be a continuos function on the interval $I$. To find how the function changes on $I$ we can find the rate of change. Let $a$ and $b$ be the endpoints of $I$ and $r$ the rate of change between those points:
$$r=\frac{f(b)-f(a)}{b-a}$$
This $r$ is the slope of the secant line and can be positive or negative.
Now, fix $a$ and move $b$ in the direction of $a$. The interval $I$ will be come more and more narrow and $f(b)$ will become more close to $f(a)$. Imagine that $|b-a|$ is very, very small, for instance $0.0000001$. The distance can be very small, but their still are two points. The human eyes can't see them.
When the $|b-a|$ is very small the secant line becomes tangent line to the graph of $f$ on $x=a$ and we can speak about the instantaneous rate of change. That is related with the speed.
The continuous rate of change or the derivative of a function F(x) is an equation f(x) which represents the the slope of F(x) at any given point. So the rate of change of a function at a specific point is just the slope of the line at that specific point.
