In calculus, everyone learns that functions are not differentiable at corners, with the absolute value function often given as a prime example.
Now consider a cow whose position vs. time function is $x(t)=|t-4|$. Obviously, the derivative of this function or the cow's velocity vs. time function is $x'(t)=\frac{t-4}{|t-4|}$. We also know that the cow's position function is not differentiable at $t=4$. From my understanding, this is because the line tangent to $x(t)$ at $t=4$ can have any slope value in between $-1$ and $1$ and is thus not unique. Mathematically, it does not seem like we can know the cow's velocity at this point.
Now thinking physically, whether it is $0$ or some other constant in between $-1$ and $1$, the cow must have a velocity at $t=4$.
My question is, how could we determine it mathematically if we cannot differentiate the cow's position function at $t=4$? Is there even enough information to mathematically determine the cow's velocity at this time? I can't really figure out what other information we need.
I know in physics, when using indefinite integrals, the constant of integration, $C$, often pops up as some initial value. I cannot figure out if this sort of thinking is applicable to my question.
Notice that the cow you are talking about changes its direction abruptly. When it approaches $0$ (distance) before time $t = 4$, it is moving in the negative direction with a constant (unit) speed, and hence the calculated velocity is $-1$. After the moment $t = 4$, i.e., for $t > 4$, the cow starts moving in the positive direction, again with a constant (unit) speed. Therefore you get the velocity $1$.
However, at the exact time $t = 4$, the cow "reverses" its direction so that velocity cannot be determined. I hope you get the essence of what problem occurs at "corners".