In mathematics, a linear approximation is an approximation of a general 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.
Definition Given a twice continuously differentiable function f {\displaystyle f} of one real variable, Taylor's theorem for the case n = 1 {\displaystyle n=1} states that
f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + R 2 {\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}}
where R 2 {\displaystyle R_{2}} is the remainder term. The linear approximation is obtained by dropping the remainder:
f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) . {\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
This is a good approximation when x {\displaystyle x} is close enough to a {\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 the tangent line to the graph of f {\displaystyle f} at ( a , f ( a ) ) {\displaystyle (a,f(a))} . 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, f ″ ( a ) {\displaystyle f''(a)} , is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the derivative is decreasing in that interval). If f {\displaystyle f} is concave up, the approximation will be an underestimate. Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function f ( x , y ) {\displaystyle f(x,y)} with real values, one can approximate f ( x , y ) {\displaystyle f(x,y)} for ( x , y ) {\displaystyle (x,y)} close to ( a , b ) {\displaystyle (a,b)} by the formula
f ( x , y ) ≈ f ( a , b ) + ∂ f ∂ x ( a , b ) ( x − a ) + ∂ f ∂ y ( a , b ) ( y − b ) . {\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).}
The right-hand side is the equation of the plane tangent to the graph of z = f ( x , y ) {\displaystyle z=f(x,y)} at ( a , b ) . {\displaystyle (a,b).}
In the more general case of Banach spaces, one has
f ( x ) ≈ f ( a ) + D f ( a ) ( x − a ) {\displaystyle f(x)\approx f(a)+Df(a)(x-a)}
where D f ( a ) {\displaystyle Df(a)} is the Fréchet derivative of f {\displaystyle f} at a {\displaystyle a} .
Applications
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. 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
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