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<title>Linear approximation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear approximation</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>linear approximation</b> is an approximation of a general <a href="Function_(mathematics)" title="Function (mathematics)">function</a> using a <a href="Linear_function" title="Linear function">linear function</a> (more precisely, an <a href="Affine_function" class="mw-redirect" title="Affine function">affine function</a>). They are widely used in the method of <a href="Finite_differences" class="mw-redirect" title="Finite differences">finite differences</a> to produce first order methods for solving or approximating solutions to equations.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Given a twice continuously differentiable function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> of one <a href="Real_number" title="Real number">real</a> variable, <a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a> for the case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

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<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

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</math></span><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"></span> states that

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<annotation encoding="application/x-tex">{\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}}</annotation>

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{2}}</annotation>

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</math></span><img src="./35f571121c264178676d1df8ab899f238a39bc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{2}}" loading="lazy"></span> is the remainder term. The linear approximation is obtained by dropping the remainder:

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<annotation encoding="application/x-tex">{\displaystyle f(x)\approx f(a)+f'(a)(x-a).}</annotation>

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</p><p>This is a good approximation when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is close enough to <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>;</span> 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 <a href="Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> to the graph of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,f(a))}">

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<annotation encoding="application/x-tex">{\displaystyle (a,f(a))}</annotation>

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</math></span><img src="./2250e602b41f737373c715600ba8f2d2cad1759d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.391ex; height:2.843ex;" alt="{\displaystyle (a,f(a))}" loading="lazy"></span>. For this reason, this process is also called the <b>tangent line approximation</b>. Linear approximations in this case are further improved when the <a href="Second_derivative" title="Second derivative">second derivative</a> of a, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(a)}">

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<annotation encoding="application/x-tex">{\displaystyle f''(a)}</annotation>

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</math></span><img src="./adfd68c9498a2589ac7cb33a1c81574b9249bf7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.497ex; height:3.009ex;" alt="{\displaystyle f''(a)}" loading="lazy"></span>, is sufficiently small (close to zero) (i.e., at or near an <a href="Inflection_point" title="Inflection point">inflection point</a>).

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is <a href="Concave_down" class="mw-redirect" title="Concave down">concave down</a> in the interval between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>, the approximation will be an overestimate (since the derivative is decreasing in that interval). If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is <a href="Concave_up" class="mw-redirect" title="Concave up">concave up</a>, the approximation will be an underestimate.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Linear approximations for <a href="Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> matrix. For example, given a differentiable function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation>

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</math></span><img src="./29473ed0c4e838ac9dbe074535e507166c0e9101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}" loading="lazy"></span> with real values, one can approximate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation>

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</math></span><img src="./29473ed0c4e838ac9dbe074535e507166c0e9101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

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</math></span><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"></span> close to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation>

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</math></span><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"></span> by the formula

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<annotation encoding="application/x-tex">{\displaystyle 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).}</annotation>

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</p><p>The right-hand side is the equation of the plane tangent to the graph of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle z=f(x,y)}</annotation>

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</math></span><img src="./1eefb2840000f404c8c0f3f5d6d72f2624854591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b).}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b).}</annotation>

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</math></span><img src="./b828bde62741f9a659d25b1264068c81c3d0b946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.717ex; height:2.843ex;" alt="{\displaystyle (a,b).}" loading="lazy"></span>

</p><p>In the more general case of <a href="Banach_space" title="Banach space">Banach spaces</a>, one has

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<annotation encoding="application/x-tex">{\displaystyle f(x)\approx f(a)+Df(a)(x-a)}</annotation>

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df(a)}">

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<annotation encoding="application/x-tex">{\displaystyle Df(a)}</annotation>

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</math></span><img src="./17c83c98b9b68b09db509888726883bae6a68aba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle Df(a)}" loading="lazy"></span> is the <a href="Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Optics">Optics</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gaussian_optics" title="Gaussian optics">Gaussian optics</a></div>

<p><i>Gaussian optics</i> is a technique in <a href="Geometrical_optics" title="Geometrical optics">geometrical optics</a> that describes the behaviour of light rays in optical systems by using the <a href="Paraxial_approximation" title="Paraxial approximation">paraxial approximation</a>, in which only rays which make small angles with the <a href="Optical_axis" title="Optical axis">optical axis</a> of the system are considered.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 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 <a href="Sphere" title="Sphere">sphere</a>. 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.

</p>

<div class="mw-heading mw-heading3"><h3 id="Period_of_oscillation">Period of oscillation</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Pendulum" title="Pendulum">Pendulum</a></div>

<p>The period of swing of a <a href="Pendulum_(mathematics)" class="mw-redirect" title="Pendulum (mathematics)">simple gravity pendulum</a> depends on its <a href="Length" title="Length">length</a>, the local <a href="Gravitational_acceleration" title="Gravitational acceleration">strength of gravity</a>, and to a small extent on the maximum <a href="Angle" title="Angle">angle</a> that the pendulum swings away from vertical, <span class="texhtml"><i>θ</i><sub>0</sub></span>, called the <a href="Amplitude" title="Amplitude">amplitude</a>.<sup id="cite_ref-Milham1945_3-0" class="reference"><a href="#cite_note-Milham1945-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It is independent of the <a href="Mass" title="Mass">mass</a> of the bob. The true period <i>T</i> 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 <a href="Pendulum_(mathematics)" class="mw-redirect" title="Pendulum (mathematics)">pendulum</a>), one example being the <a href="Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>:<sup id="cite_ref-Nelson_4-0" class="reference"><a href="#cite_note-Nelson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle T=2\pi {\sqrt {L \over g}}\left(1+{\frac {1}{16}}\theta _{0}^{2}+{\frac {11}{3072}}\theta _{0}^{4}+\cdots \right)}</annotation>

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</p><p>where <i><b>L</b></i> is the length of the pendulum and <i><b>g</b></i> is the local <a href="Gravitational_acceleration" title="Gravitational acceleration">acceleration of gravity</a>.

</p><p>However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> ) the <a href="Frequency" title="Frequency">period</a> is:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}\qquad \qquad \qquad \theta _{0}\ll 1}">

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}\qquad \qquad \qquad \theta _{0}\ll 1}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

<p>In the linear approximation, the period of swing is approximately the same for different size swings: that is, <i>the period is independent of amplitude</i>. This property, called <a href="Isochronism" class="mw-redirect" title="Isochronism">isochronism</a>, is the reason pendulums are so useful for timekeeping.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

</p>

<div class="mw-heading mw-heading3"><h3 id="Electrical_resistivity">Electrical resistivity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Electrical_resistivity" class="mw-redirect" title="Electrical resistivity">Electrical resistivity</a></div>

<p>The electrical resistivity of most materials changes with temperature. If the temperature <i>T</i> does not vary too much, a linear approximation is typically used:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (T)=\rho _{0}[1+\alpha (T-T_{0})]}</annotation>

</semantics>

</math></span></span>

where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is called the <i>temperature coefficient of resistivity</i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation>

</semantics>

</math></span><img src="./55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" loading="lazy"></span> is a fixed reference temperature (usually room temperature), and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation>

</semantics>

</math></span><img src="./d9c04a9d26b86af8c6205ba2a6287fd655b6b714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}" loading="lazy"></span> is the resistivity at temperature <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation>

</semantics>

</math></span><img src="./55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" loading="lazy"></span>. The parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is different for different reference temperatures. For this reason it is usual to specify the temperature that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> was measured at with a suffix, such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{15}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{15}}</annotation>

</semantics>

</math></span><img src="./2e93f2a04a064beebf70cf0756e0cb1c8fb5a1bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.364ex; height:2.009ex;" alt="{\displaystyle \alpha _{15}}" loading="lazy"></span>, and the relationship only holds in a range of temperatures around the reference.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.

</p>

<ul><li><a href="Binomial_approximation" title="Binomial approximation">Binomial approximation</a></li>

<li><a href="Euler's_method" class="mw-redirect" title="Euler's method">Euler's method</a></li>

<li><a href="Finite_differences" class="mw-redirect" title="Finite differences">Finite differences</a></li>

<li><a href="Finite_difference_methods" class="mw-redirect" title="Finite difference methods">Finite difference methods</a></li>

<li><a href="Newton's_method" title="Newton's method">Newton's method</a></li>

<li><a href="Power_series" title="Power series">Power series</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor series</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">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</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130303014028/http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html">"12.1 Estimating a Function Value Using the Linear Approximation"</a>. Archived from <a rel="nofollow" class="external text" href="http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html">the original</a> on 3 March 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">3 June</span> 2012</span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFLipsonLipsonLipson2010" class="citation book cs1">Lipson, A.; Lipson, S. G.; Lipson, H. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aow3o0dhyjYC&pg=PA51"><i>Optical Physics</i></a> (4th ed.). Cambridge, UK: Cambridge University Press. p. 51. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-49345-1</bdi>.</cite></span>

</li>

<li id="cite_note-Milham1945-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Milham1945_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFMilham1945" class="citation book cs1">Milham, Willis I. (1945). <i>Time and Timekeepers</i>. MacMillan. pp. <span class="nowrap">188–</span>194. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1744137">1744137</a>.</cite></span>

</li>

<li id="cite_note-Nelson-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nelson_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFNelsonM._G._Olsson1987" class="citation journal cs1">Nelson, Robert; M. G. Olsson (February 1987). <a rel="nofollow" class="external text" href="http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf">"The pendulum – Rich physics from a simple system"</a> <span class="cs1-format">(PDF)</span>. <i>American Journal of Physics</i>. <b>54</b> (2): <span class="nowrap">112–</span>121. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986AmJPh..54..112N">1986AmJPh..54..112N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.14703">10.1119/1.14703</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121907349">121907349</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2008-10-29</span></span>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFBeckettand_three_more1911" class="citation encyclopaedia cs1">Beckett, Edmund; and three more (1911). <span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Clock"><a class="external text external" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Clock">"Clock" </a></span>. In <a href="Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a> (ed.). <i><a href="Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. Vol. 06 (11th ed.). Cambridge University Press. pp. <span class="nowrap">534–</span>553, see page 538, second para. <q>Pendulum.—</q></cite> includes a derivation</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFHallidayRobert_ResnickJearl_Walker1997" class="citation book cs1">Halliday, David; Robert Resnick; Jearl Walker (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofp000davi/page/381"><i>Fundamentals of Physics, 5th Ed</i></a></span>. New York: John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofp000davi/page/381">381</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-14854-7</bdi>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFCooper2007" class="citation book cs1">Cooper, Herbert J. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=t7OoPLzXwiQC&pg=PA162"><i>Scientific Instruments</i></a>. New York: Hutchinson's. p. 162. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4067-6879-4</bdi>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFWard1971" class="citation book cs1">Ward, M. R. (1971). <i>Electrical Engineering Science</i>. McGraw-Hill. pp. <span class="nowrap">36–</span>40. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-094255-2</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFWeinstein,_AlanMarsden,_Jerrold_E.1984" class="citation book cs1">Weinstein, Alan; Marsden, Jerrold E. (1984). <i>Calculus III</i>. Berlin: Springer-Verlag. p. 775. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90985-0</bdi>.</cite></li>

<li><cite id="CITEREFStrang,_Gilbert1991" class="citation book cs1">Strang, Gilbert (1991). <i>Calculus</i>. Wellesley College. p. 94. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9614088-2-0</bdi>.</cite></li>

<li><cite id="CITEREFBock,_DavidHockett,_Shirley_O.2005" class="citation book cs1">Bock, David; Hockett, Shirley O. (2005). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780764177668/page/118"><i>How to Prepare for the AP Calculus</i></a></span>. Hauppauge, NY: Barrons Educational Series. p. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780764177668/page/118">118</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7641-2382-3</bdi>.</cite></li></ul>

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