Posts Tagged ‘mathematics’

Stationary points of f(x)=x+√x

June 4, 2017
fx

The curves in the pic are f(x) and f'(x).

Question: The function is f(x)=x+√x. Can you find its min or max value? If you calculate f'(x) and put it to 0, you get x=1/4. But draw the curve and there is no such max/min there.

One suggestion:   Clearly f'(x) has two asymptotes at x=0 and y=1 and so can never be 0 for any real value of x. But why does differentiating lead to the result x=1/4? I agree that f(x) = x-√x has a clear minimum there, but it’s a different function. My answer was that since there’s an infinity at x=0 f'(x) is undefined, but I’m not convinced that’s the answer.

OK, so √x is defined as the positive value of x^(1/2).

Let’s see if we can differentiate it:

x = √x√x  where x, √x >= 0.

D(x)  = 1 = 2√x*D(x) => D(x) = 1/(2√x)

So, f'(x) = 1 + 1/(2√x)

if f'(x) =0, then √x = -1/2.  But √x is >= 0 by definition, so there is no solution.

Let’s try to be a little more systematic while staying within the bounds of high school maths.

We consider the function  g(x,c) = x + x^(1/2)

where x^(1/2) = (√x)*(-1)^c

and we choose c to be either 0 or 1.

So g'(x) = 1 + (1/2√x)*(-1)^c

Setting g'(x) = 0 =>

√x = (-1/2)*(-1)^c

But since  √x is positive, c must be 1 rather than 0 for this to have a solution.

So by accepting a solution here we force the original function to be g(x) = x – √x.

 

Some errors in ‘Mathematics of Life’ by Ian Stewart

January 11, 2014

MATHEMATICSOFLIFE

This book was interesting overall–I thought the most engaging part was the argument about why genetic engineering is hazardous, while I was disappointed that it didn’t really address the question of why and how far maths was relevant to biology, merely giving a number of topics as examples.  There were also some alarming errors that I noticed:

More precisely, what leaks is ions:  charged atomic nuclei.  (p161)  Since we’re talking about nerve conduction, I suppose that must be Na11+–doesn’t sound too healthy…

Experience shows that continuum models are very effective provided  the discrete components are much bigger than the effects being described.  (p178)  ‘Bigger’ as in smaller, I guess.

The light green [lacewings] are more common on grass, so light green females encounter light green males; similarly for dark green insects on conifers…The end result is identical:  assortative mating can occur, opening the door to sympatric speciation.  (p236)

...assortative mating.  Organisms in a given group share similar habits, eat similar food, and therefore meet up more often than they do with members of the other group.  (p239)

That’s ‘assortative’ as in associative–the opposite.

Finally, on page 256 we have this diagram:

mathslife1_0002_NEWwhile the corresponding text on page 257 says:  Two neighbouring cells with low levels of Notch activity never occurred.  And the caption says that the white cells are the ones with low Notch activity.  Go figure.

The Use of Diagram in Greek Mathematics, British Academy 14 March

March 14, 2013
euclid

Euclid fragment from Oxyrhynchus

Addressing this topic, Reviel Netz said it was possible to use the comparative method to establish the archetype of these diagrams, although one could not be sure this archetype was the same as the original used by the ancient author.

The ancient Greek mathematical diagram was different from modern mathematical diagrams because it contained new information that was not in the text and it was schematic rather than pictorial (you could have straight lines represented by curves for instance–the thing was to get the configuration right, and it was topological rather than metric).  It differed from ancient Babylonian and Chinese texts because there the text contained instructions relating to something external that was to be gridded, cut, folded and so on while the Greek diagram was part of the text.

netz

Picture of Reviel Netz

The Greek mathematical text differed from the contemporary literary text because that was devoid of illustrations or other forms of articulation (word divisions, capitals, etc) and merely comprised one letter after another.  This was because it was merely a blueprint with which one could reconstruct a speech act.

So how had the very schematic diagram originated?  Perhaps it was by influence of the schematic uniformity of the written text, rather than some epistemological insight of the first Greek mathematicians.  Thus the diagram became part of the text, and mathematics (geometry) shared in the prestige of literary culture.

To which I would comment that a combination of images and argument is surely characteristic of mathematical thought, and the fact one didn’t bother to sort it out may merely have meant a greater closeness assumed between author and reader.  The diagram below (by Albert Einstein) after all doesn’t look very much like gravitational lensing…

lensing