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I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place?

In this moment I just know that it is named the derivative and that it is the slope of the tangent line at that point. We can find that slope by finding the limit of closer and closer to the point slopes. I still don't understand what does it really represent.

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  • $\begingroup$ Think about the velocity or the acceleration of an object. $\endgroup$
    – AlienRem
    Feb 8, 2016 at 11:33
  • $\begingroup$ What about the speed related to distance ? $\endgroup$ Feb 8, 2016 at 11:34

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Consider this example: You get into your car and drive around. The function $f(t)$ tells you how much distance you have covered until time $t$. Now the derivative of this function is your momentaneous velocity.

So, since you only asked for an interpretation, think about this: Did you ever sit in a car wondering how fast you were going 'in that instance'? Of course, technically speaking, the tachometer only gives you some average speed over the last second or so but one still often thinks of it as the speed of the car 'right now'.

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  • $\begingroup$ It is funny to see that speed is the first thing which come to our minds ! $\endgroup$ Feb 8, 2016 at 11:35
  • $\begingroup$ Indeed. I think it is just the most accessible example since everybody has experienced it in some way :). Also one could argue that 'rate of change' is basically the definition of speed. $\endgroup$
    – j4GGy
    Feb 8, 2016 at 11:36
  • $\begingroup$ Thank you, I was a little confused. Now it is clearer $\endgroup$
    – Bogdan Pop
    Feb 8, 2016 at 11:43
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The functions we are working with in calculus are (at least at the introductory level) continuous. This basically means that no matter how far in we zoom on them, they will still be smooth. This is why we can find the instantaneous rate of change at some point; we can always just look at some ever smaller interval $\Delta x$ and compute the value of the slope $\frac{\Delta y}{\Delta x}$. If the function is continuous, then we can do this for as small a $\Delta x$ as we want, so if we let $\Delta x \rightarrow 0$, we get the limit that the slope at that point tends toward and we call that $\frac{dy}{dx}$ or instantaneous rate of change.

In the real world, things are never really continuous, but often it is a fine approximation, so we can use the tools we learn from calculus anyways.

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