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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a linear approximation is an approximation of a general function using a linear function (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 generall ...
). 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 th ...
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. Mo ...
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 Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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 Kähle ...
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 vector ...
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-valued ...
of f at a.


Applications


Optics

Gaussian optics is a technique in
geometrical optics Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of '' rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
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 An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light pro ...
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 geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. 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 am ...
. 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 ele ...
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 pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
), 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 derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
*
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 ser ...


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