Tangent Plane

Just like a curve has a tangent line at a given point, a surface has an intersection of tangent lines, i.e. a tangent plane.

#todo WHY?

$$z=z_0+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$

Tangent Planes §

When graphed, the derivatives of a 2D function can be seen as the ‘direction’ of the point in both the x and y axes. Just like how a tangent line is defined as $y-y_p=f^{\prime }(x_p)\times (x-x_p)$ (Equation of a line), we can define a tangent plane using an intersection of two tangents:

Linear Approximation §

Using the tangent plane, we can obtain a linear approximation of $f$ at $P_1=(x,y)$, given we known the equation to the tangent plane at $P_0=(x_0,y_0,z_0)$ and $P_1\sim P_0$ (P1 is very close to P0):

$$f(x,y)\approx z_0+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$

Or, in different notation, where $f_x|P_0$ and $f_y|P_0$ gives the value of the partial derivatives $f_x$ and $f_y$ at point $P_0$:

$$f(x,y)\approx z_0+f_x|P_0\times (x-x_0)+f_y|P_0\times (y-y_0)$$

Approximate Change §

Using the same equation for linear approximation:

$$f(x,y)\approx z_0+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$

we can rearrange it (by moving the $z_0$):

$$f(x,y)-f(x_0,y_0)\approx f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$

$\Delta x=x-x_0$ and $\Delta y=y-y_0$. Hence, $\Delta f=\Delta x+\Delta y$ and the final equation is:

$$\Delta f=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y$$

Examples §

1: Tangent Plane Linear Approximation

Let $f(x,y)$ = $x\times e^{xy}$. Find the tangent plane to the surface $z=f(x,y)$ at the point where $(x,y)=(2,0)$. Hence approximate $f(2.1,-0.1)$

$z_0=f(2,0)=2e^{2\times 0}=2$ $f_x(x,y)=e^{xy}+yx{e}^{xy}$ $f_y(x,y)=x^2{e}^{xy}$ $f_x(2,0)=e^0+0=1$ $f_y(2,0)=4\times e^0=4$ Tangent plane: $z=2+1(x-2)+4(y-0)=2-2+x+4y=x+4y$ Now, for the linear approximation: $f(2.1,-0.1)\approx 2+1(2.1-2)+4(-0.1-0)$ $=2+0.1-0.4=1.7$ $f(2.1,-0.1)\approx 1.7$