Difference between second order derivative and curvature.

Solution 1:

The second derivative can give you an idea of how a graph is shaped, but curvature has a specific mathematical definition. It's related to the radius of curvature, which is more of a geometric concept.

The radius of curvature at a specific point is the radius of a circle that you would have to draw that would exactly match up with a curve at that point. The curvature is then defined as the inverse of the radius of curvature. So a large radius of curvature indicates a graph is nearly flat. This means the curvature, as the inverse of the radius of curvature, would be nearly zero for a line that is nearly straight. The more curled a graph is, the higher it's curvature value.

As an example, consider the simple parabola, $y=x^2$. This function has a constant second derivative of 2. This gives you an idea the graph will be concave up. The curvature will not be constant though. At each point on the parabola, you would need a different size circle if you wanted the circle to 'match up' with the graph, so each point will have a different curvature.