Inmathematics, alinear approximation is an approximation of a generalfunction using alinear function (more precisely, anaffine function). They are widely used in the method offinite differences to produce first order methods for solving or approximating solutions to equations.

Given a twice continuously differentiable function${\displaystyle f}$  of onereal variable,Taylor's theorem for the case${\displaystyle n=1}$  states that$${\displaystyle f(x)=f(a)+f^{\prime }(a)(x-a)+R_2}$$ where${\displaystyle R_2}$  is the remainder term. The linear approximation is obtained by dropping the remainder:$${\displaystyle f(x)\approx f(a)+f^{\prime }(a)(x-a).}$$ 

This is a good approximation when${\displaystyle x}$  is close enough to${\displaystyle a}$ ; 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 thetangent line to the graph of${\displaystyle f}$  at${\displaystyle (a,f(a))}$ . For this reason, this process is also called thetangent line approximation. Linear approximations in this case are further improved when thesecond derivative of a,${\displaystyle f^{″}(a)}$ , is sufficiently small (close to zero) (i.e., at or near aninflection point).

If${\displaystyle f}$  isconcave down in the interval between${\displaystyle x}$  and${\displaystyle a}$ , the approximation will be an overestimate (since the derivative is decreasing in that interval). If${\displaystyle f}$  isconcave up, the approximation will be an underestimate.^[1]

Linear approximations forvector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by theJacobian matrix. For example, given a differentiable function${\displaystyle f(x,y)}$  with real values, one can approximate${\displaystyle f(x,y)}$  for${\displaystyle (x,y)}$  close to${\displaystyle (a,b)}$  by the formula$${\displaystyle f\left(x,y\right)\approx f\left(a,b\right)+\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\left(a,b\right)\left(x-a\right)+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\left(a,b\right)\left(y-b\right).}$$ 

The right-hand side is the equation of the plane tangent to the graph of${\displaystyle z=f(x,y)}$  at${\displaystyle (a,b).}$ 

In the more general case ofBanach spaces, one has$${\displaystyle f(x)\approx f(a)+Df(a)(x-a)}$$ where${\displaystyle Df(a)}$  is theFréchet derivative of${\displaystyle f}$  at${\displaystyle a}$ .

Gaussian optics is a technique ingeometrical optics that describes the behaviour of light rays in optical systems by using theparaxial approximation, in which only rays which make small angles with theoptical axis of the system are considered.^[2] 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 asphere. 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.

The period of swing of asimple gravity pendulum depends on itslength, the localstrength of gravity, and to a small extent on the maximumangle that the pendulum swings away from vertical,θ₀, called theamplitude.^[3] It is independent of themass of the bob. The true periodT of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (seependulum), one example being theinfinite series:^[4][5]$${\displaystyle T=2\pi \sqrt{\frac{L}{g}}\left(1+\frac{1}{16}{\theta }_0^2+\frac{11}{3072}{\theta }_0^4+\cdots \right)}$$ 

whereL is the length of the pendulum andg is the localacceleration of gravity.

However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,^{[Note 1]} ) theperiod is:^[6]

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, calledisochronism, is the reason pendulums are so useful for timekeeping.^[7] Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

The electrical resistivity of most materials changes with temperature. If the temperatureT does not vary too much, a linear approximation is typically used:$${\displaystyle \rho (T)={\rho }_0[1+\alpha (T-T_0)]}$$ where${\displaystyle \alpha }$  is called thetemperature coefficient of resistivity,${\displaystyle T_0}$  is a fixed reference temperature (usually room temperature), and${\displaystyle {\rho }_0}$  is the resistivity at temperature${\displaystyle T_0}$ . The parameter${\displaystyle \alpha }$  is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation,${\displaystyle \alpha }$  is different for different reference temperatures. For this reason it is usual to specify the temperature that${\displaystyle \alpha }$  was measured at with a suffix, such as${\displaystyle {\alpha }_{15}}$ , and the relationship only holds in a range of temperatures around the reference.^[8] When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.

• Binomial approximation
• Euler's method
• Finite differences
• Finite difference methods
• Newton's method
• Power series
• Taylor series

• Weinstein, Alan; Marsden, Jerrold E. (1984).Calculus III. Berlin: Springer-Verlag. p. 775.ISBN 0-387-90985-0.
• Strang, Gilbert (1991).Calculus. Wellesley College. p. 94.ISBN 0-9614088-2-0.
• Bock, David; Hockett, Shirley O. (2005).How to Prepare for the AP Calculus. Hauppauge, NY: Barrons Educational Series. p. 118.ISBN 0-7641-2382-3.