of the bob. The true period T of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (see pendulum), one example being the infinite series: where L is the length of the pendulum and g is the local acceleration of gravity. However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings, ) the period is: In the linear approximation, the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time. Electrical resistivity The electrical resistivity of most materials changes with temperature. If the temperature T does not vary too much, a linear approximation is typically used: where is called the temperature coefficient of resistivity, is a fixed reference temperature (usually room temperature), and is the resistivity at temperature . The parameter is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, is different for different reference temperatures. For this reason it is usual to specify the temperature that was measured at with a suffix, such as , and the relationship only holds in a range of temperatures around the reference. When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used. See also Binomial approximation Euler's method Finite differences Finite difference methods Newton's method Power series Taylor series Notes References Further reading Differential calculus Numerical analysis First order methods