Approximation (2) - 선형 근사

4. Linear Approximation

 

곡선은 접점 근방에서 접선과 매우 가깝게 놓여있다.

y=f(x) is differentiable at x=a

 

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

(Linear Approximation of f at a or Tangent Line Approximation of f at a)

 

$ L(x)=f(a)+f'(a)(x-a) $

(Linearization of f at a)

 

ex) $ \sqrt{9.09} $

Let $ f(x)=\sqrt{x} $

$ f'(x)=\frac{1}{2\sqrt{x}} $

$ f(9)=3, f'(9)=\frac{1}{6} $

a=9에서의 $ L(x)=f(9)+f'(9)(x-9)=3+\frac{1}{6}(x-9)=\frac{1}{6}x+\frac{3}{2} $

$ \sqrt{x}\approx \frac{1}{6}x+\frac{3}{2} $ (when x is near 9)

$ \sqrt{9.09}=f(9.09)\approx \frac{9.09}{6}+\frac{3}{2}=3.015 $