All Topics  
Linear approximation

 

 
   Email Print
   Bookmark   Link
 

 

Linear approximation
 
 
  Topics Linear approximation Topic Home

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a linear approximation is an approximation of a general function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 using a linear function
Linear function

In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication....
 (more precisely, an affine function).

n a differentiable function f of one real
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 variable, Taylor's theorem
Taylor's theorem

In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function around a given point by polynomials whose coefficients depend only on the derivatives of the function at that point....
 for n=1 states that

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

which is true for x close to a.






Discussion
Ask a question about 'Linear approximation'
Start a new discussion about 'Linear approximation'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a linear approximation is an approximation of a general function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 using a linear function
Linear function

In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication....
 (more precisely, an affine function).

Definition

Given a differentiable function f of one real
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 variable, Taylor's theorem
Taylor's theorem

In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function around a given point by polynomials whose coefficients depend only on the derivatives of the function at that point....
 for n=1 states that

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

which is true for x close to a. The expression on the right-hand side is just the equation for the tangent line to the graph of f at (a, f(a)), and for this reason, this process is also called the tangent line approximation.

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
Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
 matrix. For example, given a differentiable function with real values, one can approximate for close to by the formula

The right-hand side is the equation of the plane tangent to the graph of at

In the more general case of Banach space
Banach space

In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them....
s, one has

where is the Fréchet derivative
Fréchet derivative

In mathematics, the Fr?chet derivative is a derivative defined on Banach spaces. Named after Maurice Fr?chet, it is commonly used to formalize the concept of the functional derivative used widely in mathematical analysis, especially functional analysis....
 of at .

Examples

To find an approximation of one can do as follows.

  1. Consider the function Hence, the problem is reduced to finding the value of .
  2. We have
  4. According to linear approximation
  6. The result, 2.926, lies fairly close to the actual value 2.924…


See also

  • Taylor expansion
  • Power series
    Power series

    In mathematics, a power series is an infinite series of the formwhere an represents the coefficient of the nth term, c is a constant, and x varies around c ....