Everything is linear to a first approximation

In, a linear approximation is an approximation of a general using a  (more precisely, an ). They are widely used in the method of to produce first order methods for solving or approximating solutions to equations.

Definition
Given a twice continuously differentiable function $$f$$ of one variable,  for the case $$n = 1 $$ states that


 * $$ f(x) = f(a) + f'(a)(x - a) + R_2\ $$

where $$R_2$$ is the remainder term. The linear approximation is obtained by dropping the remainder:


 * $$ f(x) \approx f(a) + f'(a)(x - a)$$.

This is a good approximation for $$x$$ when it is close enough to $$a$$; 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 the to the graph of $$f$$ at $$(a,f(a))$$. For this reason, this process is also called the tangent line approximation.

If $$f$$ is in the interval between $$x$$ and $$a$$, the approximation will be an overestimate (since the derivative is decreasing in that interval). If $$f$$ is, the approximation will be an underestimate.

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

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

In the more general case of s, one has


 * $$ f(x) \approx f(a) + Df(a)(x - a)$$

where $$Df(a)$$ is the of $$f$$ at $$a$$.

Optics
Gaussian optics is a technique in that describes the behaviour of light rays in optical systems by using the, in which only rays which make small angles with the  of the system are considered. 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. 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
The period of swing of a depends on its, the local , and to a small extent on the maximum  that the pendulum swings away from vertical, θ0, called the. It is independent of the 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 ), one example being the :



T = 2\pi \sqrt{L\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right) $$

where L is the length of the pendulum and g is the local.

However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings, ) the is:


 * $$T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 \qquad (1)\,$$

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

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:
 * $$\rho(T) = \rho_0[1+\alpha (T - T_0)]$$

where $$\alpha$$ is called the temperature coefficient of resistivity, $$T_0$$ is a fixed reference temperature (usually room temperature), and $$\rho_0$$ is the resistivity at temperature $$T_0$$. The parameter $$\alpha$$ is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, $$\alpha$$ is different for different reference temperatures. For this reason it is usual to specify the temperature that $$\alpha$$ was measured at with a suffix, such as $$\alpha_{15}$$, and the relationship only holds in a range of temperatures around the reference. When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.