Linearization

In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f ( x ) {displaystyle y=f(x)} at any x = a {displaystyle x=a} based on the value and slope of the function at x = b {displaystyle x=b} , given that f ( x ) {displaystyle f(x)} is differentiable on [ a , b ] {displaystyle } (or [ b , a ] {displaystyle } ) and that a {displaystyle a} is close to b {displaystyle b} . In short, linearization approximates the output of a function near x = a {displaystyle x=a} . For example, 4 = 2 {displaystyle {sqrt {4}}=2} . However, what would be a good approximation of 4.001 = 4 + .001 {displaystyle {sqrt {4.001}}={sqrt {4+.001}}} ? For any given function y = f ( x ) {displaystyle y=f(x)} , f ( x ) {displaystyle f(x)} can be approximated if it is near a known differentiable point. The most basic requisite is that L a ( a ) = f ( a ) {displaystyle L_{a}(a)=f(a)} , where L a ( x ) {displaystyle L_{a}(x)} is the linearization of f ( x ) {displaystyle f(x)} at x = a {displaystyle x=a} . The point-slope form of an equation forms an equation of a line, given a point ( H , K ) {displaystyle (H,K)} and slope M {displaystyle M} . The general form of this equation is: y − K = M ( x − H ) {displaystyle y-K=M(x-H)} . Using the point ( a , f ( a ) ) {displaystyle (a,f(a))} , L a ( x ) {displaystyle L_{a}(x)} becomes y = f ( a ) + M ( x − a ) {displaystyle y=f(a)+M(x-a)} . Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to f ( x ) {displaystyle f(x)} at x = a {displaystyle x=a} . While the concept of local linearity applies the most to points arbitrarily close to x = a {displaystyle x=a} , those relatively close work relatively well for linear approximations. The slope M {displaystyle M} should be, most accurately, the slope of the tangent line at x = a {displaystyle x=a} . Visually, the accompanying diagram shows the tangent line of f ( x ) {displaystyle f(x)} at x {displaystyle x} . At f ( x + h ) {displaystyle f(x+h)} , where h {displaystyle h} is any small positive or negative value, f ( x + h ) {displaystyle f(x+h)} is very nearly the value of the tangent line at the point ( x + h , L ( x + h ) ) {displaystyle (x+h,L(x+h))} . The final equation for the linearization of a function at x = a {displaystyle x=a} is: y = ( f ( a ) + f ′ ( a ) ( x − a ) ) {displaystyle y=(f(a)+f'(a)(x-a))}