# Using differentials to estimate a maximum error

• Jul 11th 2009, 03:36 PM
arbolis
Using differentials to estimate a maximum error
Four positive numbers lesser than 50 each are rounded to the first decimal and then multiplied between themselves. Use differentials to estimate the maximum possible error of the product.
My attempt : Let $x$, $y$ and $z$ be the 3 numbers.
We have $(x+\Delta x)(y+\Delta y)(z+\Delta z)=0.1(zx+zy+xy)+0.01(z+x+y)+0.001$.
Note that I considered $\Delta x = \Delta y = \Delta z =0.1$.
However I don't see how I can use differentials. I don't see any way to involve them. So I didn't solve the problem...
Do you have any idea?
• Jul 11th 2009, 07:14 PM
malaygoel
If P is the product and x,y,z are numbers, then

$P=xyz$

hence, we have,
$dP=yzdx + xzdy + xydz$

For maximum possible error, dx, dy and dz are maximum.

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You have used

$
\Delta x = \Delta y = \Delta z =0.1
$

It cant be >0.5
• Jul 11th 2009, 09:09 PM
arbolis
Ok you've done $dP=\frac{\partial P}{\partial x}+\frac{\partial P}{\partial y}+\frac{\partial P}{\partial z}$.
But then you wrote
Quote:

For maximum possible error, dx, dy and dz are maximum.
but these are differentials. I don't see how $dx$, $dy$ and $dz$ can have a maximum value if any numerical given value.
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Quote:
Ok, so I'm correct then?