Taylor approximations

Functions of several variables: Total differential and Taylor approximation

Taylor approximations

For a neat function $f(x)$ of one variable we have already seen that we can approximate the graph of the function near a point $x=a$ by the tangent line at this point defined by the equation

$y=f(a)+f'(a)\cdot (x-a)\tiny.$ For functions of two variables, we can define a similar linear approximation (also referred to as first-order approximation) near a point; in this case we use the tangent plane at the selected point on the graph of the given function.

For a neat function $f(x,y)$ of two variables $x$ and $y$ we can approximate function values near the point $(x,y)=(a,b)$ via the tangent plane at this point defined by the equation

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

For functions $f(x)$ of one variable we have already seen that the linear approximation was the same as the Taylor polynomial of degree 1, and that we get more precise approximations when we consider a Taylor polynomial of higher degree, for example, the following quadratic approximation (second order approximation):

$f(x)\approx f(a)+f'(a)(x-a)+\tfrac{1}{2}f''(a)(x-a)^2$ For functions of two variables we can define a similar approximation of second order:

For a neat function $f(x,y)$ of two variables $x$ and $y$ we can approximate function values near the point $(x,y)=(a,b)$ via

$\begin{aligned}f(x,y) &\approx f(a,b)+f_x(a,b){\vartriangle}x+f_y(a,b){\vartriangle}y\\ &\phantom{=}+ \frac{1}{2}\left(f_{xx}(a,b){\vartriangle}x^2+2f_{xy}(a,b){\vartriangle}x{\vartriangle}y+f_{yy}(a,b){\vartriangle}y^2\right)\end{aligned}$ with

${\vartriangle}x=x-a\quad\text{and}\quad {\vartriangle}y=y-b\tiny.$

We calculate the quadratic approximation at the point $(1,1)$ for the function

$f(x,y)=x^2y-xy^2-3xy\tiny.$ The graph of the function is as follows:

graph of the function

The required partial derivatives are

$f_x(x,y)=2xy-y^2-3y,\quad f_y(x,y)=x^2-2xy-3x,$

$f_{xx}(x,y)=2y,\quad f_{xy}(x,y)=2x-2y-3\quad\text{en}\quad f_{yy}(x,y)=-2x\tiny.$ Substitution of the coordinates of the point $(1,1)$ in the partial derivatives and in the general formula for the quadratic approximation gives

$\begin{aligned}z(x,y) &= -3-2(x-1)-4(y-1)+(x-1)^2-3(x-1)(y-1)-(y-1)^2\\ \\ &= -x+y + x^2-3xy-y^2\end{aligned}$ The graph of this approximation looks as follows:

Taylor Approach V of order 2

Together in one figure we get:

function + Taylor Approach

The function and the Taylor approximation do not differ much near $(1,1)$ ; further away though, they are substantially different.

The linear and quadratic approximations of functions of two variables originate from the Taylor's theorem of first and second order.

For a neat function $f(x,y)$ of two variables $x$ and $y$ , the following statement is true near the point $(x,y)=(a,b)$ with ${\vartriangle} x=x-a$ and ${\vartriangle}y=y-b$ :

$f(x,y) = f(a,b)+f_x(a,b){\vartriangle}x+f_y(a,b){\vartriangle}y+R_1(x,y)$ with the remainder term $R_1(x,y)$ given as

$R_1(x,y)=\frac{1}{2!}\left(f_{xx}(\xi,\eta){\vartriangle}x^2+2f_{xy}(\xi,\eta){\vartriangle}x{\vartriangle}y+f_{yy}(\xi,\eta){\vartriangle}y^2\right)$ for some $\xi$ between $a$ and $x$ , and some $\eta$ between $b$ and $y$ .

For a neat function $f(x,y)$ of two variables $x$ and $y$ , the following statement is true near the point $(x,y)=(a,b)$ with ${\vartriangle}x=x-a$ and ${\vartriangle}y=y-b$ :

$\begin{aligned}R_2(x,y) &= \frac{1}{3!}\Bigl(f_{xxx}(\xi,\eta){\vartriangle}x^3+3f_{xxy}(\xi,\eta){\vartriangle}x^2{\vartriangle}y\\ &\phantom{=xxx} +3f_{xyy}(\xi,\eta){\vartriangle}x{\vartriangle}y^2 +f_{yyy}(\xi,\eta){\vartriangle}y^3\Bigr)\end{aligned}$ for some $\xi$ between $a$ and $x$ , and some $\eta$ between $b$ and $y$ .

We could continue in this way, but we stop here. The Taylor series of of two variables $x$ and $y$ is defined as follows:

For a neat function $f(x,y)$ of two variables $x$ and $y$ , one can approximate function values near the point $(x,y)=(a,b)$ with

$\begin{aligned}f(x,y) &= f(a,b)\\ &\phantom{=}+f_x(a,b){\vartriangle}x+f_y(a,b){\vartriangle}y\\ &\phantom{=}+ \frac{1}{2!}\left(f_{xx}(a,b){\vartriangle}x^2+2f_{xy}(a,b){\vartriangle}x{\vartriangle}y+f_{yy}(a,b){\vartriangle}y^2\right)\\ &\phantom{=}+ \frac{1}{3!}\Bigl(f_{xxx}(a,b){\vartriangle}x^3+3f_{xxy}(a,b){\vartriangle}x^2{\vartriangle}y\\ &\phantom{=xxxxx}+3f_{xyy}(a,b){\vartriangle}x{\vartriangle}y^2+f_{yyy}(a,b){\vartriangle}y^3\Bigl)\\ &\phantom{=}+\cdots\end{aligned}$ with

${\vartriangle}x=x-a\quad\text{and}\quad {\vartriangle}y=y-b\tiny.$