# Linear approximation

Tangent line at (a, f(a))

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 $f$ of one real variable, Taylor's theorem for the case $n = 1$ states that

$f(x) = f(a) + f'(a)(x - a) + R_2\$

where $R_2$ is the remainder term. The linear approximation is obtained by dropping the remainder:

$f(x) \approx f(a) + f'(a)(x - a)$.

This is a good approximation for $x$ when it is close enough to $a$; 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 $f$ at $(a,f(a))$. For this reason, this process is also called the tangent line approximation.

If $f$ is concave down in the interval between $x$ and $a$, the approximation will be an overestimate (since the derivative is decreasing in that interval). If $f$ is concave up, the approximation will be an underestimate.[1]

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 $f(x, y)$ with real values, one can approximate $f(x, y)$ for $(x, y)$ close to $(a, b)$ by the formula

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

The right-hand side is the equation of the plane tangent to the graph of $z=f(x, y)$ at $(a, b).$

In the more general case of Banach spaces, one has

$f(x) \approx f(a) + Df(a)(x - a)$

where $Df(a)$ is the Fréchet derivative of $f$ at $a$.