In single variable calculus, we learned that the tangent line to the graph of $$y=f(x)$$ at $$x=x_0$$ gives a good approximation of $$f(x)$$ near $$x_0$$ if $$f'(x_0)$$ exists: $f(x)\approx L(x)=f(x_0)+f'(x_0)(x-x_0).$
Here $$L(x)$$ is called the linearization [1] of $$f$$ at the point $$x$$ (see Fig. 1).

Similarly, for 2D problems, the tangent plane to the graph of $$z=f(x,y)$$ at the point $$P=(x_0,y_0,z_0)$$ gives a good approximation to the function near $$(x_0,y_0)$$ if the function is “smooth enough” (see Fig. 2):

Definition 1. If $$z=f(x,y)$$, the function $$L(x,y)$$ which is given by the following equation is called the linearization of $$f$$ at $$(x_0,y_0)$$ $L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0).$ If $$f$$ is “smooth enough”, then for $$(x,y)$$ near $$(x_0,y_0):$$ $f(x,y)\approx L(x,y)$

Example 1

Approximate the value of $$f(0.2,-0.1)$$ given $$f(x,y)=\cos x\ e^{2y}$$.

Solution

We can use the linear approximation of $$f$$ at $$(0,0)$$, where $$f$$ and its partial derivatives can be easily evaluated: $f(0,0)=1$ \begin{aligned} f_x(x,y)=-\sin x\ e^{2y}\Rightarrow &f_x(0,0)=0\\ f_y(x,y)=2\cos x\ e^{2y}\Rightarrow &f_y(0,0)=2\end{aligned}
Therefore $$L(x,y)=1+0\times(x-0)+2\times(y-0)$$, which approximates $$f(0.2,-0.1)$$ $f(0.2,-0.1)\approx L(0,2,-0.1)=1+2\times(-0.1)=0.8$

The exact value of $$f(0.2,-0.1)=0.8024$$, which means the error in this approximation is $$\approx 0.3\%$$

### Extension to Functions of More Than Two Variables

We can easily extend the concept of the linearization of functions of more than two variables. For example, if $$u=f(x,y,z)$$, then its linearization at $$(x_0,y_0,z_0)$$ is given by:
${\small L(x,y,z)=f(x_0,y_0,z_0)+f_x(x_0,y_0,z_0)\cdot(x-x_0)+f_y(x_0,y_0,z_0)\cdot(y-y_0)+f_z(x_0,y_0,z_0)\cdot(z-z_0)}$
or ${\small L(x,y,z)=f(x_0,y_0,z_0)+\frac{\partial f}{\partial x}(x_0,y_0,z_0)\cdot(x-x_0)+\frac{\partial f}{\partial y}(x_0,y_0,z_0)\cdot(y-y_0)+\frac{\partial f}{\partial z}(x_0,y_0,z_0)\cdot(z-z_0)}$

If $$u=f(x_1,\cdots,x_n)$$, its linearization at the point $$\mathbf{p}=(a_1,\cdots, a_n)$$ is $L(x_1,\cdots,x_n)= f(\mathbf{p})+\left.\frac{\partial f}{\partial x_1}\right|_{\mathbf{p}}(x-a_1)+\cdots+\left.\frac{\partial f}{\partial x_n}\right|_{\mathbf{p}} (x-a_n)$

In the following example, we want to know whether we can approximate a function with its linearization if it is not smooth enough?

Example 2

Given $f(x,y)=\left\{\begin{array}{ll} \dfrac{xy}{x^2+y^2} & \text{if } (x,y)\neq (0,0)\\ \\ 0 & \text{if } (x,y)= (0,0) \end{array} \right.,$
using the linearization of $$f$$ at $$(0,0)$$, approximate $$f(0.01,-0.01)$$ and compare it with its exact value.

Solution

To calculate $$f_x(0,0)$$ and $$f_y(0,0)$$, we have to use the definition of partial derivatives: $f_x(0,0)=\lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h}=\lim_{h\to 0}\frac{\frac{h\times 0}{h^2+0^2}-0}{h}=\lim_{h\to 0}\frac{0-0}{h}=0,$
$f_y(0,0)=\lim_{k\to 0}\frac{f(0,0+k)-f(0,0)}{k}=\lim_{k\to 0}\frac{\frac{ 0\times k}{0^2+k^2}-0}{h}=\lim_{k\to 0}\frac{0-0}{k}=0,$
Therefore the linearization of $$f$$ at $$(0,0)$$ reads: $L(x,y)=\underbrace{f(0,0)}_{=0}+\underbrace{f_x(0,0)}_{=0} x+\underbrace{f_y(0,0)}_{=0} y=0,$
Therefore $$L(0.01,-0.01)=0$$. However the exact value of $$f(0.01,-0.01)$$ is $$-1/2$$.

In this example, $$f(x,y)=0$$ on the $$x$$ and $$y$$-axes, $$f(x,y)=\frac{1}{2}$$ on $$x=y$$ and $$f(x,y)=-\frac{1}{2}$$ on $$x=-y$$.

This example shows even if $$f_x(x_0,y_0)$$ and $$f_y(x_0,y_0)$$ exist, the linearization does not need to be a good approximation for $$f$$ near $$(x_0,y_0)$$.

[1] Here, we use “linearization” or “linear approximation” loosely. Note that $$L(x)$$ is not a linear function unless $$f(x_0)=0$$, because any linear function has to pass through the origin. More precisely we should say $$L(x)$$ is an “affine function” and the approximation is the “affine approximation”. An affine function is a function composed of a linear function + a constant.