# Using Differentials to determine maximum possible error

• Salazar
So, divide that last equation by f and you have\frac{df}{f}= \frac{dw}{w}+ \frac{dz}{z}+ \frac{dy}{y}+ \frac{dx}{x}< \frac{0.05}{30}+ \frac{0.05}{30}+ \frac{0.05}{30}+ \frac{0.05}{30}= \frac{0.05}{30}(4)= \frac{4}{600}= \frac{1}{150}= 0.00666\overline{6} < 0.007 which means that, to two decimal places, the relative error is less than 0.007

## Homework Statement

Four positive numbers, each less than 30, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding.

So our function of four variables would be : f(w,x,y,z) = wxyz
Where w,x,y,z<30

From Scratch

## The Attempt at a Solution

So I have $df = xyz\frac{\partial f}{\partial w} + wyz\frac{\partial f}{\partial x} + wxz\frac{\partial f}{\partial y} +wxy\frac{\partial f}{\partial z}$

I know that

$\frac{\partial f}{\partial w}, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \& \frac{\partial f}{\partial z} = \frac{1}{2}$

since the max error when rounding a number is .5.

My question is when solving for the error, would I substitute $w, x, y, \& z$ with 30, or 29.9.

With 30 I get 303*(.2) = 5,400.
With 29.9 I get 29.93*(.2) =5,346.17 -> and I wouldn't know where to round off.

Or is my method wrong already?

Last edited:
f=wxyz

then
df = xyz.dw + wyz.dx + wxz.dy + wxy.dz

a single partial derivative (excuse the notation) is
df/dw=xyz.1

and you know
dx,dy,dz,dw<=0.5

It might help to look at the "relative error"- that is, the error as a fraction or percentage of the actual value.
You have df= xyzdw+ xywdz+ xzwdy+ yzwdx. Dividing by the value, xyzw, gives
$$\frac{df}{xyzw}= \frac{dw}{w}+ \frac{dz}{z}+ \frac{dy}{y}+ \frac{dx}{x}$$

You know that x, y, z, and w are all less than 30 so f= xyzw< 810000. You also know that dcx, dy, dz, and dw less than 0.05.