Visualizing Second Derivative Given a graph of a simple function $f$ (continuous, no oscillation, smooth). We can roughly tell how big $f'(a)$ is for any point $a$, i.e. by looking at the steepness of the graph at that point. Is there any way to estimate $f''(a)$? Determining its sign is quite easy by checking the convexity of $f$ at that point. But how do we know how big $f''(a)$ is?
 A: Just like the size of $f'(a)$ is related to how "steep" the graph of $f(x)$ is at $a$, the size of $f''(a)$ is related to have "curved" the graph of $f(x)$ is at $a$. That is, $f''(a)$ relates to the curvature. See the picture in the Wikipedia page: the size of the second derivative is the reciprocal of the radius of the osculating circle (literally, "kissing circle") at the point $(a,f(a))$. Explicitly the curvature at the point $a$ is given by the expression
$$\frac{|f''(a)|}{(1+(f'(a))^2)^{3/2}}$$
so its size gives you information about the size of $f''(a)$ (since you already know how big $f'(a)$ is).
Alternatively, you can think about how the slope of the tangent changes as you move a little bit away from $x=a$. The faster the slope changes, the larger the second derivative is in absolute value (because it means the rate of change of $f'(x)$ is very large). If the slope changes very little when you move away from $a$, then the second derivative is small in absolute value.
A: The second derivative is related to the curvature of the graph; a larger value of the second derivative means that the slope is increasing at a faster rate, and so the function is curving upward faster. So the second derivative can be roughly interpreted as how tightly the graph is curving at that point.
It's not quite that simple though, since as the slope increases the same second derivative will create softer curves; consider the fact that a parabola has a constant second derivative, but the curve of the graph is clearly sharpest at the vertex. So you have to also take into account how steep the function is when estimating the second derivative.
Overall, it is difficult to visually estimate the second derivative since it depends on both the curvature and the steepness of a graph.
