Linear Approximation

$f(x)\approx f(a)+f^{\prime }(a)(x-a)$ is the linear approximation or tangent line approximation for x near a

$f(x,y)\approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$ is the linear approx or tangent plane approx for (x, y) close to (a, b)

Definition: The Linearization of f(x, y) at (a, b) is defined as $L_{(a,b)}(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$

Thus, the linear approximation of f at (a, b) can be written as $f(x,y)\approx L_{a,b}(x,y)$ for (x, y) close to (a, b)

Linear approximation can also be expressed as

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

or

$△f\approx f_x(a,b)△x+f_y(a,b)△y$ for $(x,y)$ near the point $(a,b)$