Sketching graph of f(x) from graph of f'(x)

[CustardTheory]

Hi all,

I understand that in the graph of f'(x), zeros and local extrema are local extrema and points of inflection, respectively, for f(x). The basic concept of checking direction and rough changes in slope to find concavity also makes sense to me.
However, I'm having a bit of a mental block trying to visualize f(x) from f'(x) when f'(x) goes off inflecting without crossing the x axis.

http://imageshack.us/a/img202/5001/captureipi.png

For example, in the above graph I understand what's happening at the first critical point and when h'(x) crosses the x axis, but I don't logically understand what's going on after the second zero. I can easily go through like a robot to find concavity via the second derivative, but I'd like to actually understand the logic behind what's happening here.

 

[bossman27]

]I can easily go through like a robot to find concavity via the second derivative, but I'd like to actually understand the logic behind what's happening here.

That essentially is the logic of what's happening here. Since the second derivative gives you concavity information, you should be able to grasp what's happening to f(x). Drawing graphs that satisfy the necessary f'(x) and f''(x) conditions should give you this intuition if you don't feel like you have it already. Just think about what happens when the concavity of the function changes while the first derivative remains positive. What should that look like?

If you want a more concretely physical way to imagine it, just think of the functions f(x), f'(x) and f''(x) representing position, velocity, and acceleration, respectively.

 

[CustardTheory]

Ah ok. It finally clicked. I think my definition of concavity was just off.
I thought it needed to be a full U shape to be counted as concave up or concave down. Am I correct in saying that this doesn't need to be the case?

 

[bossman27]

Ah ok. It finally clicked. I think my definition of concavity was just off.
I thought it needed to be a full U shape to be counted as concave up or concave down. Am I correct in saying that this doesn't need to be the case?

Yes that is correct, one way to visualize it might be that positive or negative concavity at a point defines whether the function is "curving up" or "curving down," relative to the tangent line at that point. Whether the tangent line sits "below" or "on top" of the function, so to speak. At point where concavity changes (f''(x) = 0), the tangent line will be on top of the function to one side of x, and below on the other.

Quantitatively, of course, it's just a measure of the instantaneous rate of change of the slope of f(x).