# Big O notation (for calculus, not computer science)

1. Jul 21, 2010

### kungal

I understand the formal definition for big O notation but is there an intuitive interpretation?. For example, if

Code (Text):
f(x) = O(x[SUP]-1/4[/SUP])
is it reasonable to say that for large n f(x) grows at the same rate as
Code (Text):
n[SUP]-1/4[/SUP]
?

2. Jul 21, 2010

### arildno

An O-notation requires a limit to which the independent variable is supposed to go.

3. Jul 21, 2010

### kungal

The O's should actually be Op's (my mistake) and x is a set of random variables. I don't think this changes the nature of the question drastically.

Note that the result f(x) = Op(x-1/4) is quite common so it can't be a meaningless statement.

4. Jul 21, 2010

### arildno

Yes it is.

Is it meant that f(x) is O(x^(-1/4) ) as x goes to zero, or as x goes to, say, infinity?
That is two entirely different situations, and needs, therefore, to be specified.
Hence, the meaninglessness of your first line.

5. Jul 21, 2010

### HallsofIvy

I hate to disagree with Arildno but it is common practice (though perhaps "abuse of notation") to use just O(f(x)) to mean "as x goes to infinity".

6. Jul 21, 2010

### arildno

Well, if that is the general default notation, I'll make a note of that.

In my own applied maths books, they dutifully make explicit what limiting operation we are speaking about

7. Jul 21, 2010

### kungal

I'm glad we've cleared that up but is it reasonable to say that
if f = Op(n-1/4) then for large n f grows at the same rate as n-1/4?

8. Jul 21, 2010

### arildno

Not at all.

f might, for example, become more and more strongly oscillatory as x goes to infinty, even though f's magnitude will be bounded by some constant multiplied by x^(-1/4).

For example, let
$$f(x)=Ax^{-\frac{1}{4}}\cos(x^{2})$$

This f is definitely O(x^(-1/4)), but its rate of growth will, be:
$$\frac{df}{dx}\to{-2A}x^{\frac{3}{4}}\sin(x^{2}), x\to\infty$$

9. Jul 21, 2010

### kungal

Thanks

10. Jul 22, 2010

### aprillove20

Well, O's should actually be Op's (my mistake) and x is a set of random variables.
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