Let us assume that all functions are continuous. I was teaching my calculus students the other day. We were talking about what points of non-differentiability look like. Two ways a function can fail to be differentiable at a point is if it looks like $y=|x|$ or like a Brownian motion (think of $x\sin x$ for instance), where the derivative oscillates too much. However, I do not have an intuition about $C^1$ functions and how they differ from $C^i$ functions for higher $i$. An example that I know is the function $$f(x)=x^2,x\geq 0\mbox{ and }f(x)=-x^2,x\leq 0.$$ The graph of this actually looks smooth to me. So the question rephrased may be:
how can one visually tell the difference between $C^1$ functions and $C^2$ functions in a straight forward way.
Although this is for undergrads, I wouldn't mind a more advanced answer.