Approximation of functions, linear methods

Approximation of Functions

Value of a function can be approximated to a very precise extent by linear approximation.This is primarily done by taking a straight line tangent to the curve a x=a.This approximation holds good for values that are sufficiently close to a itself.Consider the function:-

 $y=f(x)$

Let the derivative of $f(x)$ be:-

 $y'={\frac {df(x)}{dx}}$

Therefore tangent to the curve of f(x) at x=a is given by:-

$(y-f(a))={\frac {df(a)}{dx}}(x-a)$
$y=f(a)+{\frac {df(a)}{dx}}(x-a)$

File:Approximation.png

Putting x=a+$\Delta x$
$y\approx f(a)+{\frac {df(a)}{dx}}(\Delta x)$

Now as $\Delta x\to 0$, $y\to f(a)$