경희대 박종도 ¶

먼저 2D일 때를 알아본다.
접선,tangent_line 식은

$\displaystyle y=f(a)+f'(a)(x-a)$

임을 상기.

Approximation to $\displaystyle y=f(x)$ near $\displaystyle a$

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

: the linear approximation of $\displaystyle f$ at $\displaystyle a.$

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

: the linearization of $\displaystyle f$ at $\displaystyle a.$

$\displaystyle f(a)$ 는 쉽게 계산 가능.

$\displaystyle L(a) = f(a)$
$\displaystyle L(x) \approx f(x)$ when $\displaystyle x$ is near $\displaystyle a$ (x가 a 근방에선 값이 비슷하다)

그리고 이제 3D(multivariable case)를 알아본다.
The tangent plane to $\displaystyle z=f(x,y)$ at $\displaystyle (a,b,f(a,b))$ is

$\displaystyle z=f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$

where $\displaystyle f_x$ and $\displaystyle f_y$ are continuous.

Linearization:

$\displaystyle L(x,y) := f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$

Linear approximation or tangent plane approximation:

$\displaystyle f(x,y)\approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$

Red Ruby Wiki