How to calculate the maximum absolute relative error using linear approximation?

Asked Oct 18 '19 at 15:42

Active Oct 18 '19 at 16:42

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Here is the problem

Using linear approximation, determine the maximum absolute relative error for the function:

$f(x,y,z) = \frac{−4⋅x^3⋅z}{y^3}$

at (1,3,2), assuming that the relative errors with respect to x, y and z are at most 0.8%, 0.3% and 1.2%, respectively. Give your answer as a percentage.

What I came up with:

So I calculated the partial derivatives with respect to (x,y,z) and came up with the linear approximation equation:

$L(x,y,z) = -\frac{8}{27}-\frac{8}{9}*(x-1)-\frac{4}{27}*(y-3)+\frac{8}{27}*(z-2)$

Then I evaluated L(1.008, 3.009, 2.024) and I got $-\frac{2009}{6750}$

Then I compared this value to the value of $f(1, 3, 2)$ and got the answer of 1.0045% which wasn't correct though. What am I doing wrong here?

edited Oct 18 '19 at 16:42

asked Oct 18 '19 at 15:42

MatiasC

1

You forgot the constant, that is, $f(1,3,2)$. – Jean-Claude Arbaut Oct 18 '19 at 15:54
Oh! You're right! Such a dumb error :D. Thanks! – MatiasC Oct 18 '19 at 15:55
You must also take care that all coefficient are not of the same sign. To get the max error you need to make sure they add up. – Jean-Claude Arbaut Oct 18 '19 at 15:58
Um not quite sure what you mean by that? Could you elaborate please – MatiasC Oct 18 '19 at 16:06
If all of $x,y,z$ are slightly above resp. $1,3,2$, then the first two terms of $L$ are negative and the third is positive, so the errors compensate. You want the maximum relative error, the worst case scenario, that is when the errors don't compensate. – Jean-Claude Arbaut Oct 18 '19 at 16:06
I now edited my question with the correct equation. I just still don't get the coefficent thing. What should I change in the equation? I think it is correct. – MatiasC Oct 18 '19 at 16:47
Yes, the equation is correct. But the error you compute is not the maximum possible given the constraints. What happens if $x>1,y>3$ but $z<2$? – Jean-Claude Arbaut Oct 18 '19 at 16:51
The error is larger isn't it? – MatiasC Oct 18 '19 at 17:04
Right. ${}{}{}{}$ – Jean-Claude Arbaut Oct 18 '19 at 17:09
Thank you now I understand! – MatiasC Oct 18 '19 at 17:14

How to calculate the maximum absolute relative error using linear approximation?

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