3. Tangent Plane and Linearization

Tangent Line

Equation

$y=f(x_0)+f^{\prime }(x_0)(x-x_0)$

3D Representation

When $y=y_0$

$z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)$

When $x=x_0$

$z=f(x_0,y_0)+f_y(x_0,y_0)(y-y_0)$

Tangent Plane

$a(z-f(x_0,y_0))+b(x-x_0)+c(y-y_0)=0$

into

$z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

Linearization - One Variable

Info

Tangent Line: $y=f(a)+f^{\prime }(a)(x-a)$ where $L(a)=f(a)$

Linearization: $L(x)=f(a)+f^{\prime }(a)(x-a)$ where $L^{\prime }(a)=f^{\prime }(a)$

Linearization - Two Variables

Info

Tangent Line:

$z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

Linearization:

$L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

where

$L(x_0,y_0)=f(x_0,y_0)$

$L_x(x_0,y_0)=f_x(x_0,y_0)$

$L_y(x_0,y_0)=f_y(x_0,y_0)$

Linearization

Definition

The linearization of a function $f(x_1,\dots ,x_n)$ at the point $(a_1,\dots ,a_n)$ is the unique linear function $L(x_1,\dots ,x_n)$ such that

$L(a_1,\dots ,a_n)=f(a_1,\dots ,a_n)$

and

$L_{x_i}(a_1,\dots ,a_n)=f_{x_i}(a_1,\dots ,a_n)$

for all $i=1,\dots ,n$.

Theorem

$L(x_1,\dots ,x_n)=f(a_1,\dots ,a_n)+{\sum }_{i=1}^n{f}_{x_i}(a_1,\dots ,a_n)(x_i-a_i)$

Higher Order

Definition

The nth order Taylor polynomial of $f$ at the point $a$ is

$T_n(x)={\sum }_{k=0}^n\frac{1}{k!}{f}^{(k)}(a)(x-a{)}^k$

Note: A similar notion for functions of several variables exists, e.g.

$T_2(x,y)=L(x,y)+\frac{1}{2}{f}_{xx}(x_0,y_0)(x-x_0{)}^2$

$+f_{xy}(x_0,y_0)(x-x_0)(y-y_0)+\frac{1}{2}{f}_{yy}(x_0,y_0)(y-y_0{)}^2$

Differential

Definition

Give a function $z=f(x,y)$ the differential $dz=df$ of $f$ is given by

$dz=df=f_x(x,y)dx+f_y(x,y)dy$

Note: If we set $x=x_0$, $y=y_0$, $dx=△x$, and $dy=△y$ we obtain the approximation

$△z=f(x_0+△x,y_0+△y)-f(x_0,y_0)\approx L(x_0+△x,y_0+△y)-L(x_0,y_0)=dz$