Editing Linear approximation

{{short description|Approximation of a function by its tangent line at a point}}
[[File:TangentGraphic2.svg|thumb|300px|Tangent line at (''a'', ''f''(''a''))]]
In [[mathematics]], a '''linear approximation''' is an approximation of a general [[function (mathematics)|function]] using a [[linear function]] (more precisely, an [[affine function]]). 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 <math>f</math> of one [[real number|real]] variable, [[Taylor's theorem]] for the case <math>n = 1 </math> states that
<math display="block"> f(x) = f(a) + f'(a)(x - a) + R_2 </math>
where <math>R_2</math> is the remainder term. The linear approximation is obtained by dropping the remainder:
<math display="block"> f(x) \approx f(a) + f'(a)(x - a).</math>

This is a good approximation when <math>x</math> is close enough to {{nowrap|<math>a</math>;}} 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 <math>f</math> at <math>(a,f(a))</math>. For this reason, this process is also called the '''tangent line approximation'''. Linear approximations in this case are further improved when the [[second derivative]] of a, <math> f''(a) </math>, is sufficiently small (close to zero) (i.e., at or near an [[inflection point]]).

If <math>f</math> is [[concave down]] in the interval between <math>x</math> and <math>a</math>, the approximation will be an overestimate (since the derivative is decreasing in that interval). If <math>f</math> is [[concave up]], the approximation will be an underestimate.<ref>{{cite web |title=12.1 Estimating a Function Value Using the Linear Approximation |url=http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html |access-date=3 June 2012 |archive-date=3 March 2013 |archive-url=https://web.archive.org/web/20130303014028/http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html |url-status=dead }}</ref>

Linear approximations for [[vector (geometric)|vector]] functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the [[Jacobian matrix and determinant|Jacobian]] matrix. For example, given a differentiable function <math>f(x, y)</math> with real values, one can approximate <math>f(x, y)</math> for <math>(x, y)</math> close to <math>(a, b)</math> by the formula 
<math display="block">f\left(x,y\right)\approx f\left(a,b\right) + \frac{\partial f}{\partial x} \left(a,b\right)\left(x-a\right) + \frac{\partial f}{\partial y} \left(a,b\right)\left(y-b\right).</math>

The right-hand side is the equation of the plane tangent to the graph of <math>z=f(x, y)</math> at <math>(a, b).</math>

In the more general case of [[Banach space]]s, one has
<math display="block"> f(x) \approx f(a) + Df(a)(x - a)</math>
where <math>Df(a)</math> is the [[Fréchet derivative]] of <math>f</math> at <math>a</math>.

==Applications==

===Optics===
{{main|Gaussian 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 which only rays which make small angles with the [[optical axis]] of the system are considered.<ref>{{cite book |first1=A. |last1=Lipson |first2=S. G. |last2=Lipson |first3=H. |last3=Lipson |url=https://books.google.com/books?id=aow3o0dhyjYC&pg=PA51 |title=Optical Physics |edition=4th |year=2010 |publisher=Cambridge University Press |location=Cambridge, UK |page=51 |isbn=978-0-521-49345-1 }}</ref> 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.

===Period of oscillation===
{{main|Pendulum}}
The period of swing of a [[Pendulum (mathematics)#Simple gravity pendulum|simple gravity pendulum]] depends on its [[length]], the local [[Gravitational acceleration|strength of gravity]], and to a small extent on the maximum [[angle]] that the pendulum swings away from vertical, {{math|''θ''<sub>0</sub>}}, called the [[amplitude]].<ref name="Milham1945">{{cite book |last=Milham |first=Willis I. |title=Time and Timekeepers |year=1945 |publisher=MacMillan |pages=188–194 |oclc=1744137 }}</ref> It is independent of the [[mass]] 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 (mathematics)|pendulum]]), one example being the [[infinite series]]:<ref name="Nelson">{{cite journal | last = Nelson | first = Robert |author2=M. G. Olsson | title = The pendulum – Rich physics from a simple system | journal = American Journal of Physics | volume = 54 | issue = 2 | pages = 112–121 | date = February 1987 | url = http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf | doi = 10.1119/1.14703 | access-date = 2008-10-29 | bibcode = 1986AmJPh..54..112N | s2cid = 121907349 }}</ref><ref>{{Cite EB1911|wstitle= Clock |volume= 06 |last1= Beckett |first1= Edmund |last2= and three more| pages = 534&ndash;553; see page 538, second para |quote= Pendulum.—}} includes a derivation</ref>
<math display="block">
T = 2\pi \sqrt{L\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right)
</math>

where '''''L''''' is the length of the pendulum and '''''g''''' is the local [[Gravitational acceleration|acceleration of gravity]].

However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,<ref group = Note>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</ref> ) the [[Frequency|period]] is:<ref>{{cite book |last = Halliday
 |first      = David
 |author2    = Robert Resnick
 |author3    = Jearl Walker
 |title      = Fundamentals of Physics, 5th Ed.
 |publisher  = John Wiley & Sons.
 |year       = 1997
 |location   = New York
 |page       = [https://archive.org/details/fundamentalsofp000davi/page/381 381]
 |url        = https://archive.org/details/fundamentalsofp000davi/page/381
 |url-access = registration
 |isbn       = 0-471-14854-7
}}</ref>
{{NumBlk||<math display="block">T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 </math>|{{EquationRef|1}}}}

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.<ref>{{cite book
  | last = Cooper
  | first = Herbert J.
  | title = Scientific Instruments
  | publisher = Hutchinson's
  | year = 2007
  | location = New York
  | page = 162
  | url = https://books.google.com/books?id=t7OoPLzXwiQC&pg=PA162
  | isbn = 978-1-4067-6879-4}}</ref> Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

===Electrical resistivity===
{{main|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:
<math display="block">\rho(T) = \rho_0[1+\alpha (T - T_0)]</math>
where <math>\alpha</math> is called the ''temperature coefficient of resistivity'', <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>\rho_0</math> is the resistivity at temperature <math>T_0</math>. The parameter <math>\alpha</math> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <math>\alpha</math> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math>\alpha</math> was measured at with a suffix, such as <math>\alpha_{15}</math>, and the relationship only holds in a range of temperatures around the reference.<ref>{{cite book |first=M. R. |last=Ward |year=1971 |title=Electrical Engineering Science |pages=36–40 |publisher=McGraw-Hill |isbn=0-07-094255-2 }}</ref> 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==
{{reflist|group=Note}}

==References==
{{reflist}}

==Further reading==
*{{cite book |author1=Weinstein, Alan |author2=Marsden, Jerrold E. |title=Calculus III |publisher=Springer-Verlag |location=Berlin |year=1984|isbn=0-387-90985-0 |page= 775}}
* {{cite book |author=Strang, Gilbert |title=Calculus |publisher=Wellesley College |year=1991|isbn=0-9614088-2-0 |page= 94}}
*{{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 |page=[https://archive.org/details/isbn_9780764177668/page/118 118] |url-access=registration |url=https://archive.org/details/isbn_9780764177668/page/118 }}

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[[Category:Differential calculus]]
[[Category:Numerical analysis]]
[[Category:First order methods]]