Editing Curvature (section)

===Graph of a function===
The [[graph of a function]] {{math|''y'' {{=}} ''f''(''x'')}}, is a special case of a parametrized curve, of the form
:<math>\begin{align}
x&=t\\
y&=f(t).
\end{align}</math>
As the first and second derivatives of {{mvar|x}} are 1 and 0, previous formulas simplify to 
:<math>\kappa = \frac{\left|y''\right|}{\bigl(1+{y'}^2\bigr)\vphantom{'}^{3/2}},</math>
for the curvature, and to
:<math>k = \frac{y''}{\bigl(1+{y'}^2\bigr)\vphantom{'}^{3/2}},</math>
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 {{mvar|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 {{mvar|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 
:<math>k(x)=y'' \Bigl(1 + O\bigl({\textstyle y'}^2\bigr) \Bigr).</math>
 
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 system|nonlinear]] to be treated approximately as linear.