Recently, I have realized how much I have taken for granted in understanding the slope of a line and the slope of a curve in general. With that said, I wanted to clear up my understanding to make sure I am on the right track.
After reading this question: "Understanding the slope of a line as a rate of change", I see now how the slope of the line $y = f(x): = mx + b$ (where $f: \mathbb{R} \to \mathbb{R}$) is the rate of change of $y$ with respect to $x$. However, where I get a little confused is this: since the value $y$ of the function changes by an amount $\Delta y$ for every $\Delta x$ change in $x$, therefore $y$ changes by an amount $ m =\frac{\Delta y}{\Delta x}$. So, if we were to study the line $y = \frac{2}{3}x$, does that mean the slope of the line can be interpreted as "two units of $y$ every three units of $x$"? With more relatable units, if $y$ is measured in meters and $x$ is measured in seconds, would the rate of change of a particle traveling along this line be read as "two meters every three seconds"?
This seems right, however in general for any differentiable curve $g: \mathbb{R} \to \mathbb{R}$ that may not be a line i've seen for example that $g'(3) = \frac{2}{5} $ (given in meters/seconds) is read as "two-fifths meters per second"; simply because the derivative is the rate of change at a point (or at an instant).
Is my understanding correct at all?
Best Answer
Yes, you are exactly right. And if it helps you build a more physical intuition, the units of the slope are the units of $y$ divided by the units of $x$. So the number of hot-dogs Takeru Kobayashi eats in a hot-dog eating competition can be modeled with the line $y = m x$, where $y$ has units of hot dogs, $x$ has units of minutes, and $m \simeq 6.6$ hot dogs / minute. Very impressive.
As for more general differential curves on any domain: a slope is a slope. The fact that a derivative exists guarantees that $g(x)$ can be approximated as a line in the neighborhood centered at $x = 3$. It is possible to overthink this. Why not eat a hot dog?