TITLE: If instantaneous rates of change aren't that rigorous, how correct is the usage of instantaneous rates of change (like velocity) by physicists?
QUESTION [6 upvotes]: According to this answer, instantaneous rates of change are more intuitive than they are rigorous.
I tend to agree with that answer because, in the Wikipedia article on differential calculus, they aren't defining the derivative to be the slope at a particular point. They define it as, "The derivative of a function at a chosen input value describes the rate of change of the function near that input value." Although this isn't wrong, the definition has been written rather safely, and I think that was intentional. They didn't define it as the slope of the graph at a particular point. It is only in the explanations section of the Derivative wiki article that they did that: "The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x. If x and y are real numbers, and if the graph of f is plotted against x, derivative is the slope of this graph at each point."
So, are physicists using terms like "instantaneous velocity" merely from an intuitive standpoint? What is the physical significance of instantaneous rates of change?

REPLY [7 votes]: I think you failed to realize that "slope of $f$ at $x$" is meaningless until you define it, but how can you define it precisely? That is the real problem.
You cannot just say "tangent", either, because that just swaps the problem for a worse problem. Not only is it difficult to define "tangent", it fails on cases like $f : ℝ→ℝ$ defined by $f(0) = 0$ and $f(x) = x^2·\cos(1/x)$ for every $x∈ℝ_{≠0}$, if you try to define "tangent" in terms of a line that touches only once in a small open disk around the point.
Ultimately, one of the best ways to define "slope/gradient of $f$ at $x$" is via the standard definition. You might ask "Why that definition?", and the answer is:

Because that definition gives nice properties, and even corresponds with intuition!

See, suppose you have a smooth hillside, and you happen to know a function $f$ that captures its elevation for every coordinate along a straight path up that hill. Then the rigorous definition of $f'$ actually gives you something that coincides with the intuitive notion of slope! The reason is that slope is intuitively "how fast it goes up", and you cannot measure it at a single point but you can measure it near a point. If you zoom in on that point on the hillside and it gradually looks more and more straight as you zoom in, then it is (rigorously provable to be) differentiable there and indeed $f'$ at that point is exactly the slope of the line that you approach as you zoom in.
Similarly, the instantaneous speed of a vehicle cannot really be measured in real life, but what the speedometer measures is how many rounds the wheel turns over a small time interval. In fact, you should notice that if you jerk the accelerator pedal on and off rapidly, the speedometer actually fails to show you the rapidly changing speed, precisely because it does not measure instantaneous speed and its time resolution is not that good. But if you travel on a relatively smooth journey, then the mathematically rigorous derivative matches closely with what the speedometer shows, because each small part of the speed-time graph is close to linear.