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In mathematics, a linear approximation is an approximation of a general function using a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
(more precisely, an
affine function In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More genera ...
). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.


Definition

Given a twice continuously differentiable function f of one real variable,
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the ...
for the case n = 1 states that f(x) = f(a) + f'(a)(x - a) + R_2 where R_2 is the remainder term. The linear approximation is obtained by dropping the remainder: f(x) \approx f(a) + f'(a)(x - a). This is a good approximation when x 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the graph of f at (a,f(a)). For this reason, this process is also called the tangent line approximation. If f is concave down in the interval between x and a, the approximation will be an overestimate (since the derivative is decreasing in that interval). If f 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 In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant *Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler m ...
matrix. For example, given a differentiable function f(x, y) with real values, one can approximate f(x, y) for (x, y) close to (a, b) by the formula f\left(x,y\right)\approx f\left(a,b\right) + \frac \left(a,b\right)\left(x-a\right) + \frac \left(a,b\right)\left(y-b\right). The right-hand side is the equation of the plane tangent to the graph of z=f(x, y) at (a, b). In the more general case of
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
s, one has f(x) \approx f(a) + Df(a)(x - a) where Df(a) is the
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-value ...
of f at a.


Applications


Optics

Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the
paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the opti ...
, 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 A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. 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.


Period of oscillation

The period of swing of a simple gravity pendulum depends on its
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
, the local strength of gravity, and to a small extent on the maximum
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
that the pendulum swings away from vertical, , called the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
. It is independent of the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
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 A pendulum is a weight suspended from a wikt:pivot, pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, Mechanical equilibrium, equilibrium position, it is subject to a restoring force due to gravity that ...
), one example being the infinite series: T = 2\pi \sqrt \left( 1+ \frac\theta_0^2 + \frac\theta_0^4 + \cdots \right) 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,A "small" swing is one in which the angle θ is small enough that sin(θ) can be approximated by θ when θ is measured in radians ) 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: \rho(T) = \rho_0 +\alpha (T - T_0)/math> where \alpha is called the ''temperature coefficient of resistivity'', T_0 is a fixed reference temperature (usually room temperature), and \rho_0 is the resistivity at temperature T_0. The parameter \alpha is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, \alpha is different for different reference temperatures. For this reason it is usual to specify the temperature that \alpha was measured at with a suffix, such as \alpha_, 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 In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial dom ...
*
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real ...
*
Power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
*
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...


Notes


References


Further reading

* * *{{cite book , author1=Bock, David , author2=Hockett, Shirley O. , title=How to Prepare for the AP Calculus , publisher=Barrons Educational Series , location=Hauppauge, NY , year=2005 , isbn=0-7641-2382-3 , pag
118
, url-access=registration , url=https://archive.org/details/isbn_9780764177668/page/118 Differential calculus Numerical analysis First order methods