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Linearization of a function

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 ${\displaystyle y=f(x)}$ at any ${\displaystyle x=a}$ based on the value and slope of the function at ${\displaystyle x=b}$, given that ${\displaystyle f(x)}$ is differentiable on ${\displaystyle [a,b]}$ (or ${\displaystyle [b,a]}$) and that ${\displaystyle a}$ is close to ${\displaystyle b}$. In short, linearization approximates the output of a function near ${\displaystyle x=a}$.
For example, ${\displaystyle {\sqrt {4}}=2}$. However, what would be a good approximation of ${\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}}$?
For any given function ${\displaystyle y=f(x)}$, ${\displaystyle f(x)}$ can be approximated if it is near a known differentiable point. The most basic requisite is that ${\displaystyle L_{a}(a)=f(a)}$, where ${\displaystyle L_{a}(x)}$ is the linearization of ${\displaystyle f(x)}$ at ${\displaystyle x=a}$. The point-slope form of an equation forms an equation of a line, given a point ${\displaystyle (H,K)}$ and slope ${\displaystyle M}$. The general form of this equation is: ${\displaystyle y-K=M(x-H)}$.
Using the point ${\displaystyle (a,f(a))}$, ${\displaystyle L_{a}(x)}$ becomes ${\displaystyle y=f(a)+M(x-a)}$. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to ${\displaystyle f(x)}$ at ${\displaystyle x=a}$.
While the concept of local linearity applies the most to points arbitrarily close to ${\displaystyle x=a}$, those relatively close work relatively well for linear approximations. The slope ${\displaystyle M}$ should be, most accurately, the slope of the tangent line at ${\displaystyle x=a}$.

 An approximation of f(x)=x^2 at (x, f(x))

Visually, the accompanying diagram shows the tangent line of f(x) at x. At f(x+h), where h is any small positive or negative value, f(x+h) is very nearly the value of the tangent line at the point (x+h,L(x+h)).

The final equation for the linearization of a function at x=a is:

{\displaystyle y=(f(a)+f'(a)(x-a))\,}

For x=a, f(a)=f(x). The derivative of f(x) is f'(x), and the slope of f(x) at a is f'(a).
Example

To find {\sqrt {4.001}}, we can use the fact that {\sqrt {4}}=2. The linearization of f(x)={\sqrt {x}} at x=a is y={\sqrt {a}}+{\frac {1}{2{\sqrt {a}}}}(x-a), because the function f'(x)={\frac {1}{2{\sqrt {x}}}} defines the slope of the function f(x)={\sqrt {x}} at x. Substituting in a=4, the linearization at 4 is y=2+{\frac {x-4}{4}}. In this case x=4.001, so {\sqrt {4.001}} is approximately 2+{\frac {4.001-4}{4}}=2.00025. The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent.
Linearization of a multivariable function

The equation for the linearization of a function f(x,y) at a point p(a,b) is:

f(x,y)\approx f(a,b)+\left.{{\frac {{\partial f(x,y)}}{{\partial x}}}}\right|_{{a,b}}(x-a)+\left.{{\frac {{\partial f(x,y)}}{{\partial y}}}}\right|_{{a,b}}(y-b)

The general equation for the linearization of a multivariable function f(\mathbf {x} ) at a point \mathbf {p} is:

f({{\mathbf {x}}})\approx f({{\mathbf {p}}})+\left.{\nabla f}\right|_{{{\mathbf {p}}}}\cdot ({{\mathbf {x}}}-{{\mathbf {p}}})

where \mathbf {x} is the vector of variables, and \mathbf {p} is the linearization point of interest .[2]
Uses of linearization

Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation

    {\frac {d{\mathbf {x}}}{dt}}={\mathbf {F}}({\mathbf {x}},t),

the linearized system can be written as

    {\frac {d{\mathbf {x}}}{dt}}\approx {\mathbf {F}}({\mathbf {x_{0}}},t)+D{\mathbf {F}}({\mathbf {x_{0}}},t)\cdot ({\mathbf {x}}-{\mathbf {x_{0}}})

where {\mathbf {x_{0}}} is the point of interest and D{\mathbf {F}}({\mathbf {x_{0}}}) is the Jacobian of {\mathbf {F}}({\mathbf {x}}) evaluated at {\mathbf {x_{0}}}.
Stability analysis

In stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of linearization theorem. For time-varying systems, the linearization requires additional justification.[3]
Microeconomics

In microeconomics, decision rules may be approximated under the state-space approach to linearization.[4] Under this approach, the Euler equations of the utility maximization problem are linearized around the stationary steady state.[4] A unique solution to the resulting system of dynamic equations then is found.[4]
Optimization

In Mathematical optimization, cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm. The optimized result is reached much more efficiently and is deterministic as a global optimum.
See also

    Linear stability
    Tangent stiffness matrix
    Stability derivatives
    Linearization theorem
    Taylor approximation
    Functional equation (L-function)

References

The linearization problem in complex dimension one dynamical systems at Scholarpedia
Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering
G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron effects, International Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007, pp. 1079-1107

    Moffatt, Mike. (2008) About.com State-Space Approach Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.