Can the "Instantaneous rate of change" ever be an exact value?

Question

I was taught in my University's Calculus 1 course that the derivative of a function at a point is its "Instantaneous rate of change" , it was defined using this limit:

$lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

Now what is currently confusing me, is how the value of this limit can be the exact value of an "Instantaneous rate of change", shouldn't it be just a (very) good approximation ? because when I think about it, an exact "Instantaneous rate of change" does not make sense.

Wouldn't this also be the case with a tangent line's gradient at a point, it would also be an approximation, because an exact tangent line gradient wouldn't make sense because you need to have 2 points to obtain a line's gradient.

So is the way I am thinking about this correct ? or am I misunderstanding something ?

The move from average to instantaneous, or secant line to tangent line, or putting the $\lim_{h\to0}$ out in front is the "big idea" you should take away from the first part of any calculus course. If you're still in the course, go talk to the instructor, otherwise revisit the material in some other fashion. — yoyo, Oct 16 '21 at 15:58
Do these three Answers collectively answer your question?: The slope at a point requires two points? and "instantaneous rate of change" and Is instantaneous rate of change an exact value? — ryang, Oct 16 '21 at 16:27
Can it "ever" be an exact value? There is the case of a straight line, in which the difference quotient gives the same slope everywhere, i.e. "an exact value". — hardmath, Oct 16 '21 at 16:40
The way to make sense of the phrase “instantaneous rate of change” is to define it to be that limit. — Hans Lundmark, Oct 16 '21 at 17:40
@RyanG So the derivative would be the best approximation for a tangent line's slope or instantaneous rate of change of a point ? Not the exact value correct ? — Mohamed Shereef, Oct 16 '21 at 18:23
If you have actually clicked on those links, then I suggest reading them slower and giving them time to be digested. — ryang, Oct 16 '21 at 18:25

Can the "Instantaneous rate of change" ever be an exact value?

0 Answers0