If $s(t)$ is the position on the $x$-axis of a particle at time $t$, then what does the quantity $(s(t + h) - s(t))/h$ represent?
Is it instantaneous velocity? I don't think it is.
Thanks for any help! Sorry if it is too easy; I just can't remember where this comes from.
On
Remember the definition of derivative:
$$f'(t)=\frac{df}{dt}(t)=\lim_{h\to0}\,\frac{f(t+h)-f(t)}{h}$$
The $\lim$ is important here. If $h$ is not small, then the quantity is simply the average speed along the particle's trajectory -- that is, the speed $v$ that, if maintained constant by the particle along its path, would take the particle from $s(t)$ to $s(t+h)$ in the same amount of time $($which is, of course, $h)$. What this also means is that
The instantenous velocity is the limit of the average velocity as the time interval goes to $0$.
With your notation, $\frac{s(t+h)-s(t)}{h}$ is the average rate of change of the $x$ coordinate of the particle in the interval of time $(t,t+h)$, which is the same as the average of the $x$ component of the (instantaneous) velocity of the particle in that time interval. If $h\to 0$ this quantity becomes the $x$ component of the instantaneous velocity of the particle at time $t$, usually denoted by $v_x(t)$. This is just one of the three Cartesian components of the instantaneous velocity, so in general it is not the same as the instantaneous velocity. In one dimensional motion along the $x$ axis, you could identify the instantaneous velocity $\vec v$ with its only component $v_x$, although technically there are no the same, one is a vector the other is a component: $\vec v =v_x\hat i$. For three dimensional motion you will need also the $y$ and $z$ components of the velocity to get the instantaneous velocity.
Most people will use $x(t)$ for the position of the particle in the $x$ axis, and some will use $s(t)$ for the total distance traveled by the particle since $t=0$ along its trajectory (not necessarily the $x$ axis). In that case $\frac{s(t+h)-s(t)}{h}$ is the average rate at which distance is being traveled, and its limit as $h\to0$ the instantaneous speed $v(t)$ of the particle (do not confuse it with the instantaneous velocity $\vec v(t)$ of the particle, which is a vector, $v=|\vec v|$). Note that in general $v_x$ is not the same as $v$ even in one dimensional motion, in which case $v=|v_x|$. The component $v_x$ can be negative, but $v$ is always non-negative.