This has been the question of the week for me and my maths teachers.
It started off when I was emptying the bins in the playground when I heard some some advanced higher maths students talking, I heard, "you know the easy stuff like differentiation...?"
"So you think that differentiation is easy do you?" I asked. "Differentiate x to the power of x."
They did what people usually do when they're first confronted with this; completely miss that this is a function rather than a function of a function. I pointed out their errors and they giggled at me.
Still that got me thinking, what is 00 ? Do we want to say 1?
We agreed not to google this, or use our graphing calculators.So we were left to our own mathematics devices.
I saw two fixed points but when I messed around I saw that I was being stupid. The only fixed point is 1, I thought that there was another around three-quarters, things got a wee bit odd there.
The graph is special play a wee bit for yourselves...
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Oh by the way
We're assuming a Real domain and a Real co-domain. Which straight away leads us to the question, what is -5-5 ? Not Real I'd say. But what about -2-2 ? That looks a bit Real? Is there something discontinous going on in the negative Reals?
Whatever, let's do the differentiation that started this...
Let
y = xx
taking logs of both sides
ln(y) = xln(x)
And as y is a function of x, and x is a function of y we can do this
1/y(y`) = ln(x) + 1
I know, I know! I hate that diddling too but this one works. So
y` = y(ln(x) +1)
And as y = xx
We have
f`(x) = xxln(x) + xx
We can check that by integrating
Actually better we have a graph with a minima, so we can check there. I leave that as an exercise for the reader.
nellie
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I looked @ f`(x) = xxln(x) + xx and had a moment, how could that ever be 0?
And then I saw, in a moment of mathematical clarity, that ln(x) could be negative.
You do all this stuff without much thought, this learning, this TMA trail. And yet, one morning you wake up with a bunch of strange skills. You wake up in love with this world of ours and its many possibilities.
For you saw it anew.