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{{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.
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==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> 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:
This is a good approximation
▲:<math> f(x) \approx f(a) + f'(a)(x - a)</math>.
▲This is a good approximation for <math>x</math> when it is close enough to <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'''.
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 |
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
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> 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>.
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===Optics===
{{main|Gaussian optics}}
===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
<math display="block">
| ___location =▼
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
|author2 = Robert Resnick
|author3 = Jearl Walker
▲ | title = Fundamentals of Physics, 5th Ed.
▲ | publisher = John Wiley & Sons.
▲ | year = 1997
|page = [https://archive.org/details/fundamentalsofp000davi/page/381 381]
|url = https://archive.org/details/fundamentalsofp000davi/page/381
}}</ref>
▲ | isbn = 0-471-14854-7}}</ref>
▲:<math>T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 \qquad (1)\,</math>
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
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| ___location = New York
| page = 162
| url =
| isbn = 978-1-4067-6879-
▲ | isbn = 1-4067-6879-0}}</ref> Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
===Electrical
{{main|Electrical resistivity}}
The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a
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
==See also==
* [[Binomial approximation]]
* [[Euler's method]]
* [[Finite differences]]
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==Notes==
{{
==References==
{{reflist}}
*{{cite book |author=Weinstein, Alan; Marsden, Jerrold E. |title=Calculus III |publisher=Springer-Verlag |___location=Berlin |year=1984|isbn=0-387-90985-0 |oclc= |doi= |page= 775}}▼
==Further reading==
*{{cite book |
▲* {{cite book |author=
*{{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 }}
{{Authority control}}
[[Category:Differential calculus]]
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