Linear approximation: Difference between revisions

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<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>

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===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 |firstfirst1=A. |lastlast1=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''₀}}, called the [[amplitude]].<ref name="Milham1945">{{cite book |last=Milham |first=Willis I. |title=Time and Timekeepers |year=1945 |publisher=MacMillan |pages=188-194188–194 |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)

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| ___location = New York

| page = 162

| url = https://wwwbooks.google.com/books?id=t7OoPLzXwiQC&pg=PA162

| isbn = 978-1-4067-6879-04}}</ref> Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

===Electrical resistivity===

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* {{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 }}

{{Authority control}}

[[Category:Differential calculus]]