Linear Approximation

The transcendental functions we have adopted to describe natural phenomena are very difficult to evaluate, except at certain points (e.g., zero). Similarly, many complex behaviours have the property that they are simple at certain nominal operating points.

Linear approximation is a method for approximating the behaviour of a complex function by a simple function, in the neighbourhood of an operating point.

Linear Approximation of f(x) at x=a

We call this tangent line the linear approximation or the linearization of $f(x)$ at $x=a$ and denote it by $L_a(x)$:

$L_a(x)=f(a)+f^{\prime }(a)(x-a)$

• This is just the 1st order Taylor Series!

Linear Approximation for Multivariate Functions

The simple equation that you can think of on a formula sheet. $f\approx f_0+f_{x}dx+f_{y}dy$

If you expand it out, you have $f(x,y)\approx f(x_0,y_0)+f_x(x_0,y_0)\cdot (x-x_0)+f_y(x_0,y_0)\cdot (y-y_0)$

This is derived from the equation of the tangent plane of a multivariate function $z=z_0+f_x(x_0,y_0)\cdot (x-x_0)+f_y(x_0,y_0)\cdot (y-y_0)$