1 Approximations using partial derivatives

1.1 Functions of two variables

We saw in HELM booklet  16.5 how to expand a function of a single variable $f\left(x\right)$ in a Taylor series:

$f\left(x\right)=f\left(x_0\right)+\left(x-x_0\right)f^{\prime }\left(x_0\right)+\frac{{\left(x-x_0\right)}^2}{2!}{f}^{″}\left(x_0\right)+\dots$

This can be written in the following alternative form (by replacing $x-x_0$ by $h$ so that $x=x_0+h$ ):

$f\left(x_0+h\right)=f\left(x_0\right)+h{f}^{\prime }\left(x_0\right)+\frac{h^2}{2!}{f}^{″}\left(x_0\right)+\dots$

This expansion can be generalised to functions of two or more variables:

$f\left(x_0+h,y_0+k\right)\≃f\left(x_0,y_0\right)+h{f}_x\left(x_0,y_0\right)+k{f}_y\left(x_0,y_0\right)$

where, assuming $h$ and $k$ to be small, we have ignored higher-order terms involving powers of $h$ and $k$ . We define $\delta f$ to be the change in $f\left(x,y\right)$ resulting from small changes to $x_0$ and $y_0$ , denoted by $h$ and $k$ respectively. Thus:

$\delta f=f\left(x_0+h,y_0+k\right)-f\left(x_0,y_0\right)$

and so $\delta f\≃h{f}_x\left(x_0,y_0\right)+k{f}_y\left(x_0,y_0\right)$ . Using the notation $\delta x$ and $\delta y$ instead of $h$ and $k$ for small increments in $x$ and $y$ respectively we may write

$\delta f\≃\delta x.f_x\left(x_0,y_0\right)+\delta y.f_y\left(x_0,y_0\right)$

Finally, using the more common notation for partial derivatives, we write

$\delta f\≃\frac{\partial f}{\partial x}\delta x+\frac{\partial f}{\partial y}\delta y.$

Informally, the term $\delta f$ is referred to as the absolute error in $f\left(x,y\right)$ resulting from errors $\delta x,\delta y$ in the variables $x$ and $y$ respectively. Other measures of error are used. For example, the relative error in a variable $f$ is defined as $\frac{\delta f}{f}$ and the percentage relative error is $\frac{\delta f}{f}\times 100$ .

Note that to determine the error numerically we need to know not only the actual values of $\delta x$ and $\delta y$ but also the values of $x$ and $y$ at the point of interest.

Example 12

Estimate the absolute error for the function $f\left(x,y\right)=x^2+x^{3}y$

Solution

$f_x=2x+3x^{2}y;f_y=x^3.$

Then $\delta f\≃\left(2x+3x^{2}y\right)\delta x+x^3\delta y$

Task!

Estimate the absolute error for $f\left(x,y\right)=x^2{y}^2+x+y$ at the point $\left(-1,2\right)$ if $\delta x=0.1$ and $\delta y=0.025$ . Compare the estimate with the exact value of the error.

First find $f_x$ and $f_y$ :

Answer

Now obtain an expression for the absolute error:

Answer

Now obtain the estimated value of the absolute error at the point of interest:

Answer

Answer

1.2 Functions of three or more variables

If $f$ is a function of several variables $x,y,u,v,\dots$ the error induced in $f$ as a result of making small errors $\delta x,\delta y,\delta u,\delta v\dots$ in $x,y,u,v,\dots$ is found by a simple generalisation of the expression for two variables given above:

$\delta f\≃\frac{\partial f}{\partial x}\delta x+\frac{\partial f}{\partial y}\delta y+\frac{\partial f}{\partial u}\delta u+\frac{\partial f}{\partial v}\delta v+\dots$

Example 13

Suppose that the area of triangle $ABC$ is to be calculated by measuring two sides and the included angle. Call the sides $b$ and $c$ and the angle $A$ .

Then the area $S$ of the triangle is given by $S=\frac{1}{2}bcsinA.$

Now suppose that the side $b$ is measured as 4.00 m, $c$ as 3.00 m and $A$ as $30^o$ . Suppose also that the measurements of the sides could be in error by as much as $\pm 0.005$ m and of the angle by $\pm 0.01^o$ . Calculate the likely maximum error induced in $S$ as a result of the errors in the sides and angle.

Solution

Here $S$ is a function of three variables $b,c,A$ . We calculate $S=\frac{1}{2}\times 4\times 3\times \frac{1}{2}=3{\text{m}}^2.$

Now $\frac{\partial S}{\partial b}=\frac{1}{2}csinA$ , $\frac{\partial S}{\partial c}=\frac{1}{2}bsinA$ and $\frac{\partial S}{\partial A}=\frac{1}{2}bccosA$ , so

$\delta S\≃\frac{\partial S}{\partial b}\delta b+\frac{\partial S}{\partial c}\delta c+\frac{\partial S}{\partial A}\delta A=\frac{1}{2}csinA\delta b+\frac{1}{2}bsinA\delta c+\frac{1}{2}bccosA\delta A.$

Here ${\left|\delta b\right|}_{max}={\left|\delta c\right|}_{max}=0.005$ and ${\left|\delta A\right|}_{max}=\frac{\pi }{180}\times 0.01$ ( $A$ must be measured in radians). Substituting these values we see that the maximum error in the calculated value of $S$ is given by the approximation

Hence the estimated value of $S$ is in error by up to about $\pm 0.01{\text{m}}^2$ .