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Re: Calculus Query
Absolutely brilliant, Manoel! This is what I love about First. When phrontist in Virginia can put out a terrifically interesting question from Virginia, and have answers come in right on point from the UK and Brazil – well, we didn’t have anything like that going for us when I was in high school.
Manoel, however did you think of approaching the problem this way? You show extraordinary mathematical insight in choosing the right form for f(x). And phrontist – what made you think up this question? It’s absolutely first rate – you’ll do just fine at MIT when you get there.
For those who struggle with these concepts like I do, the inverse of a function exactly “undoes” what the original function did to the variable. If you perform a ‘function operation’ on a variable x, then follow up with the ‘inverse function operation’, you get the original x back.
So, if f(x) = 0.743 * x^1.618, and we take the first derivative,
then f’(x) = 0.743 * 1.618 * x^0.618 = 1.202 * x^0.618.
But, if f’(x) = Inv-f(x), we can substitute f(x) into the inverse function and get the original x back :
Inv-f(x) = 1.202 * ( f(x) )^0.618 =? x
Inv-f(x) = 1.202 * [ 0.743 * x^1.618] ^ 0.618 = 1.202 * 0.743^0.618 * (x^1.618)^0.618
= 1.202 * 0.832 * x = x, just what we started with.
So, the derivative of f(x) is the same as the inverse of f(x), just as Manoel showed us.
If we keep differentiating – which we can do as long as x >= 0, we can learn some interesting things about these functions, slopes, inflection points, concavity, etc.
f(x) = 0.743 * x^1.618
f’(x) = 1.202 * x^0.618
f’’(x) = 0.743 * x^-0.382
f’’’(x) = -0.284 * x^-1.382
From x=0 to infinity, the function f(x) increases continuously, as does the first derivative/inverse. But f(x) approaches infinity much faster than the derivative.
From f’(x) we can see that the slope of the function f(x) approaches infinity as x gets larger and larger. Eventually, f(x) is heading almost straight up.
From f ’’(x) we can see that the function is everywhere concave upwards, as f’’(x) is positive for all x>0. But f ’’(x) is the slope of the first derivative/inverse, so we can see that the inverse always has a positive slope. But f ’’(x) goes towards zero as x gets larger and larger, so the slope of the inverse goes to zero, i.e., the inverse function approaches a horizontal line. Not an asymptote, however – the inverse function can be made as large as desired by taking large enough values of x. There’s no upper limit for it, so no asymptote.
From f ’’’(x), which is always negative for all x>0, we can see that the inverse is always concave downwards.
(Manoel – you might want to re-check your graph of these functions. It’s always a good idea to double-check computer generated graphs.
Note that the function is numerically equal to its first derivative/inverse when x = 1.618.
A magical number, to be sure. More interesting properties than can be addressed here.
I haven't had this much fun since the last First competition.
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Leo M.
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