Using Differentials to determine maximum possible error

In summary: 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
  • #1
Salazar
8
0

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

Homework Equations



From Scratch

The Attempt at a Solution



So I have [itex] 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}[/itex]

I know that

[itex]\frac{\partial f}{\partial w}, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \& \frac{\partial f}{\partial z} = \frac{1}{2}[/itex]

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

My question is when solving for the error, would I substitute [itex]w, x, y, \& z[/itex] 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:
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  • #2
your differential isn't quite right
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
 
  • #3
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
[tex]\frac{df}{xyzw}= \frac{dw}{w}+ \frac{dz}{z}+ \frac{dy}{y}+ \frac{dx}{x}[/tex]

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.
 

1. What is the purpose of using differentials to determine maximum possible error?

The purpose of using differentials is to estimate the maximum possible error in a measurement or calculation. This can help scientists determine the accuracy and precision of their data and make informed decisions about the reliability of their results.

2. How do you calculate the maximum possible error using differentials?

To calculate the maximum possible error, you would first find the derivative of the function representing the measurement or calculation. Then, you would plug in the value of the independent variable (usually the measured quantity) and multiply it by the desired error tolerance.

3. Can differentials be used to determine the minimum possible error?

No, differentials can only be used to determine the maximum possible error. This is because the derivative only provides an upper bound for the error, and it is possible for the actual error to be smaller than the calculated maximum.

4. What factors can affect the accuracy of using differentials to determine maximum possible error?

The accuracy of using differentials to determine maximum possible error can be affected by the precision of the measured quantities, the complexity of the function being differentiated, and any rounding errors in the calculations.

5. Are there any limitations to using differentials for determining maximum possible error?

Yes, there are some limitations to using differentials for determining maximum possible error. This method assumes that the function is continuous and differentiable, and that the error tolerance is small enough to be negligible. It may also not account for any systematic errors in the measurement or calculation process.

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