Question about d, d^2

A-R-Q · **Joined:** Sat, 20 Nov 2010 21:17:02 UTC **Posts:** 432

This is actually from an optimization problem about maximizing distances.

With the distance formula, I have seen that instead of people using d = square root of [ (X2 - X1)^2 + (Y2 - Y1)^2], they do this:

d^2 = (X2 - X1)^2 + (Y2 - Y1)^2
^ So they square both sides, and then they say that let S = d^2, meaning let this be the distance. Why is this the same as the first?

Shadow · **Posted:** Wed, 7 Mar 2012 03:52:18 UTC

A-R-Q wrote:

This is actually from an optimization problem about maximizing distances.

With the distance formula, I have seen that instead of people using d = square root of [ (X2 - X1)^2 + (Y2 - Y1)^2], they do this:

d^2 = (X2 - X1)^2 + (Y2 - Y1)^2
^ So they square both sides, and then they say that let S = d^2, meaning let this be the distance. Why is this the same as the first?

It's not the same function, but maximizing and minimizing it is the same process. The reason they do it is because it's easier to take derivatives of polynomials than things with square roots, and the reason it is justified is because for $0\le a, b$ you know that $a\le b\iff a^2\le b^2$ , so if something is the biggest, and you square all the stuff you have, it is still the biggest. Similarly for the smallest.

outermeasure · **Posted:** Wed, 7 Mar 2012 04:46:27 UTC

Shadow wrote:

A-R-Q wrote:

This is actually from an optimization problem about maximizing distances.

With the distance formula, I have seen that instead of people using d = square root of [ (X2 - X1)^2 + (Y2 - Y1)^2], they do this:

d^2 = (X2 - X1)^2 + (Y2 - Y1)^2
^ So they square both sides, and then they say that let S = d^2, meaning let this be the distance. Why is this the same as the first?

It's not the same function, but maximizing and minimizing it is the same process. The reason they do it is because it's easier to take derivatives of polynomials than things with square roots, and the reason it is justified is because for $0\le a, b$ you know that $a\le b\iff a^2\le b^2$ , so if something is the biggest, and you square all the stuff you have, it is still the biggest. Similarly for the smallest.

Also, because d^2

is differentiable everywhere, but d isn't (it has "infinite" derivative where it vanishes).

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