Given a twice continuously differentiable function of one
Taylor's theorem for the case states that
where is the remainder term. The linear approximation is obtained by dropping the remainder:
This is a good approximation when is close enough to ; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the
tangent line to the graph of at . For this reason, this process is also called the tangent line approximation.
concave down in the interval between and , the approximation will be an overestimate (since the derivative is decreasing in that interval). If is
concave up, the approximation will be an underestimate.
Linear approximations for
vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the
Jacobian matrix. For example, given a differentiable function with real values, one can approximate for close to by the formula
The right-hand side is the equation of the plane tangent to the graph of at
Gaussian optics is a technique in
geometrical optics that describes the behaviour of light rays in optical systems by using the
paraxial approximation, in which only rays which make small angles with the
optical axis of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a
sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,[Note 1] ) the
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.
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.
^Beckett, Edmund; and three more (1911). "Clock" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 06 (11th ed.). Cambridge University Press. pp. 534–553, see page 538, second para. Pendulum.— includes a derivation