Meaning of the word 'instantaneous'

[Burnerjack]

[Moderator's note: Moved from another forum and slightly edited.]

While teaching myself Calculus, I ran into a concept that is both simple and perplexing (to me, anyway).
While I understand slope (m) as being X2-X1/Y2-Y1. Simple enough. But then there is talk of instantaneous slope. As if the difference between X1 and X2 or Y1 and Y2 are so small as to be nonexistent. I find this impossible. I have a similar issue with the tangent of a point as well, for basically the same reason. I am not looking for the mathematical explanation but rather the logical one. In a nutshell: How can a point have measurable slope or tangent? Sure would appreciate it if someone can explain this in 'everyday' terms. My mind, she is, how you say "boggled".

 

[PeterDonis]

How can a point have measurable slope or tangent?

It doesn't. What you are calling "instantaneous slope" is a derivative, and requires the concept of a limit to understand properly. The math experts in this forum can probably give much more detail and rigor than I can, but here is a brief summary:

Suppose we pick some point P on a curve, and then we look at the slope of a straight line from P to some other point Q further along the curve. That slope will be, in the notation you are using (but corrected by switching X and Y):

$$
S = \frac{Y_Q - Y_P}{X_Q - X_P}
$$

Now let's rewrite this in the functional notation that is more commonly used in this connection. The curve we are talking about is some function $y = f(x)$. Let's call $X_P$ just $x$, and let's use $h$ for the difference $X_Q - X_P$. Then we note that $Y_Q = f(x + h)$ and $Y_P = f(x)$ and we have:

$$
S = \frac{f(x+h) - f(x)}{h}
$$

Now we imagine moving point Q closer and closer to P, which corresponds to making $h$ smaller and smaller. We will find that the slope $S$ will change, but it will not change all over the place; it will approach some particular value. That value is what you are calling the "instantaneous slope" of the curve at point P, but which is more usually called the "derivative" of the function $f$ at the point $x$. In other words, we have

$$
\frac{df}{dx} = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}
$$

What does this look like on the graph of the curve we were looking at? It is the slope of a straight line that is tangent to the curve at point P. If you think about moving point Q closer and closer to point P, until finally it coincides with point P, and how the straight lines from P to Q will behave as you do this, you should be able to convince yourself that that is what you will end up with.

 

[pixel]

Another way to look at it is to magnify the area immediately around the point in question. As you magnify more and more, the neighborhood of the point will look more and more like a straight line, the same way that the surface of the Earth looks flat where we're standing. That straight line in the neighborhood of the point is the tangent to the curve at that point.

 

[epenguin]

If I am not mistaken, Great Minds were stuck on this for a long time. From what I have read giants such as Newton, Liebniz, Bernoullis, Euler didn't really have a theoretical answer, but they kind of knew what they were doing for practical purposes, and that it was only in the 19th century that it was all formulated in the way that nearly everyone now uses and learns with a doctrine of limits. But I think it could be done in a different way and there are indeed still Great Minds who think it should all be done in a different way. But if you are just learning calculus you had better go with the crowd.

 

[Ssnow]

The ratio $\frac{Y_{Q}-Y_{P}}{X_{Q}-X_{P}}$ is the "average" slope between $P$ and $Q$. If you want the instantaneous slope in a precise point $S$ between $P$ and $Q$ you must to look to the ratio $\frac{Y_{Q}-Y_{S}}{X_{Q}-X_{S}}$ when $Q \rightarrow S$. In this last case when $Q$ tends to $S$ the average slope tends to the instantaneous slope in $S$ that is the coefficient of the tangent line in $S$ (you will see that this is the derivative in the point $S$).

 

[robphy]

Another way to look at it is to magnify the area immediately around the point in question. As you magnify more and more, the neighborhood of the point will look more and more like a straight line, the same way that the surface of the Earth looks flat where we're standing. That straight line in the neighborhood of the point is the tangent to the curve at that point.

That is the argument I give to my students.
My new favorite way to show this is to use https://www.desmos.com/calculator .
Enter a function like y=(x-1)*(x-3), then use the mouse and wheel to zoom into a point of interest.
You could then add the equation of the line through that point with the slope you determine, then zoom back out.
If you want, you could make it much fancier.