Re: Positive/Negative after taking the square root
- From: Taras_96 <taras.di@xxxxxxxxx>
- Date: Sun, 9 Dec 2007 06:11:02 -0800 (PST)
So my question wasn't stupid, it seems to have generated a few
responses :).
There were a few replies which mentioned definite integrals (and thus
a region over which to integrate). Although these responses were very
interesting, the original question dealth with indefinite integrals.
eg: for int(x^2dx), you can't say whether x > 0 or x < 0, x is in R
However, it does seem like the general consensus is that if you are
making a substitution where the sign is ambiguous, you must treat each
value separately, as suggested by matt:
When you substitute u = x^2 you're losing information about which of
the two possible values of x you're dealing with. (This always happens
when the inverse function is multi-valued.) Strictly speaking you need
to keep track of the separate cases, so that everything is uniquely
defined, and then it all should come right:
I think this is similar tact taken when solving problems where it's
easier to break up the problem into smaller problems, achieved by
considering, for example, different domains of the problem, then
combining all the answers to obtain the final answer.
eg: draw |x^2 - 4|
|x^2 - 4| = |x+2||x-2|
so for x < -2, |x+2| = -(x+2), |x-2| = -(x-2), so f(x) = x^2-4
for -2<x<2, |x+2| = (x+2), |x-2| = -(x-2), so f(x) = -(x^2-4)
for x>2, |x+2| = (x+2), |x-2| = (x-2), so f(x) = x^2-4
Then combine the solutions
The difference in this case is that you would expect both of the
solutions to be the same for x < 0 and x>0 (which they are)
To avoid this ambiguity, could you possibly say let u be the variable
defined by the relationship x = sqrt(u), and thus x^2 = u? (ie: have a
1-1 mapping from x to u)
The error was in the very last step.
-1/3 (x^2)^(3/2) = -1/3 x^3 is incorrect if, as was stipulated, x <= 0.
Correctly, under that condition, we have -1/3 (x^2)^(3/2) = 1/3 x^3, and so
there is no paradox.
I'm guessing this is because if x < 0, then sqrt(x^2) = -x. This is a
bit suprising, as I've never come across it in my years of working
with exponents, nor do I ever remember treating cases separately (I
would have expected to have come across it by now!). For example, it's
taught in schools that if (x^a)^b, then x^(ab). Never when using this
rule have I considered if x > 0 or x < 0. So, using this rule,
x(^2)^1/2 = x, which doesn't agree with what I had previously. Could
it possibly be because x^0.5 != sqrt(x), as sqrt(x) by convention
implies the POSITIVE square root of x, whereas x^0.5 doesn't have this
extra condition?
One: The expression u^2 = v does NOT imply u = +/- sqrt(v). This abuse of notation doesn't really even mean >
anything mathematically, and it only serves to create ambiguity. What it DOES imply is |u| = sqrt(v) or >
equivalently sgn(u)*u = sqrt(v).
Could you explain this a bit more? It kind of makes sense, but it's a
bit hazy.
Two: You cannot make a substitution u for x which has u' = 0 for some x; in general if you make such a
substitution the result is not valid. This subtle condition may or may not be covered in your calculus class, but
the reasoning behind it will be clear when you take analysis.
This seems like a pretty restrictive condition - is there a reason why
it isn't taught at the elementary level, as it seems like a pretty
important rule
Thanks everyone for the discussion so far!
.
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