Linear approximation
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In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
For example, given a differentiable function f of one real variable, Taylor's theorem for n=1 states that
where R2 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.
One can also use linear approximations for vector functions of a vector variable, in which case the derivative at a point is replaced by the Jacobian 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
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 Banach spaces, one has
where Df(a) is the Fréchet derivative of f at a.
[edit] Examples
To find an approximation of ![\sqrt[3]{25}](../../../math/6/1/0/6103bcd9bd4b52da8971871123af4e9c.png)
one can do as follows.
- Consider the function
Hence, the problem is reduced to finding the value of f(25). - We have
- According to linear approximation
- The result, 2.926, lies fairly close to the actual value 2.924…
