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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?

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    The second derivative tells you how fast the derivative is increasing/decreasing. So, maybe you could look in a small interval around the point and see if the function appears to be getting more and more steep... based on the sign you already know. The more it seems to be changing, the bigger the absolute value.2011-11-07
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    The exact value is not obvious to see on the graph, but you may think of $f''(a)$ as a measure of the curvature of the graph at that point. The curvier the graph, the bigger $|f''(a)|$ (the sign is given by convexity as you say).2011-11-07

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