linguistic:voice_change, linguistic:paraphrasing
Graph of a function

The graph of a function y = f (x), is a special case of a parametrized curve, of the form

As the first and second derivatives of x are 1 and 0, previous formulas simplify to

for the curvature, and to

for the signed curvature.

In the general case of a curve, the sign of the signed curvature is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values of x. This makes significant the sign of the signed curvature.

The sign of the signed curvature is the same as the sign of the second derivative of f. If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. If it is zero, then one has an inflection point or an undulation point.

When the slope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, using big O notation, one has

It is common in physics and engineering to approximate the curvature with the second derivative, for example, in beam theory or for deriving the wave equation of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwise nonlinear to be treated approximately as linear.