# Linear approximation

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Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).

Definition

Given a differentiable function "f" of one real variable, Taylor's theorem for "n"=1 states that

:$f\left(x\right) = f\left(a\right) + f \text{'}\left(a\right)\left(x - a\right) + R_2$

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

:$f\left(x\right) approx f\left(a\right) + f \text{'}\left(a\right)\left(x - a\right)$

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 matrix. For example, given a differentiable function $f\left(x, y\right)$ with real values, one can approximate $f\left(x, y\right)$ for $\left(x, y\right)$ close to $\left(a, b\right)$ by the formula:$fleft\left(x,y ight\right)approx fleft\left(a,b ight\right)+frac\left\{partial f\right\}\left\{partial x\right\}left\left(a,b ight\right)left\left(x-a ight\right)+frac\left\{partial f\right\}\left\{partial y\right\}left\left(a,b ight\right)left\left(y-b ight\right).$

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

In the more general case of Banach spaces, one has

:$f\left(x\right) approx f\left(a\right) + Df\left(a\right)\left(x - a\right)$

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

Examples

To find an approximation of $sqrt \left[3\right] \left\{25\right\}$ one can do as follows.

# Consider the function $f\left(x\right)= x^\left\{1/3\right\}.,$ Hence, the problem is reduced to finding the value of $f\left(25\right)$.
# We have
#:$f \text{'}\left(x\right)= 1/3x^\left\{-2/3\right\}.$
# According to linear approximation
#:$f\left(25\right) approx f\left(27\right) + f \text{'}\left(27\right)\left(25 - 27\right) = 3 - 2/27.$
# The result, 2.926, lies fairly close to the actual value 2.924&hellip;

References

*cite book |author=Weinstein, Alan; Marsden, Jerrold E. |title=Calculus III |publisher=Springer-Verlag |location=Berlin |year=1984 |pages= page 775|isbn=0-387-90985-0 |oclc= |doi=
*
*cite book |author=Bock, David; Hockett, Shirley O. |title=How to Prepare for the AP Calculus|publisher=Barrons Educational Series |location=Hauppauge, NY |year=2005 |pages=page 118 |isbn=0-7641-2382-3 |oclc= |doi=

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