My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
November 6th, 2010, 11:42 AM   #1
 
Joined: Sep 2010

Posts: 73
Thanks: 0

determine pairs of functions independent

Reviewing for an exam, so I'll probably be posting a lot of questions. I'll try my best to figure it out myself but when I've spent too much time on it, I'll come here. So, first question:

Determine whether the pairs of functions are linearly independent or dependent on the real line.
f(x) = x^2
g(x) = x^2 * |x|

I need to take the Wronskian... but how do I do the derivative of an absolute value?
mbradar2 is offline  
 
November 6th, 2010, 12:51 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 4,025
Thanks: 78

Re: determine pairs of functions independent

Let f(x) = |x|.
f'(x)=1, x > 0
f'(x)= -1, x <0
f'(0) does not exist.
mathman is offline  
November 6th, 2010, 01:54 PM   #3
 
Joined: Sep 2010

Posts: 73
Thanks: 0

Re: determine pairs of functions independent

Well then, how do I know whether I use 1 or -1 for f'(x)?

Either way, I get the result does not equal zero, so they are linearly dependent. What if I come across an absolute value where f'(x) = 1 would make it dependent and f'(x) = -1 would make it independent? Then what?
mbradar2 is offline  
November 7th, 2010, 12:20 AM   #4
 
Joined: Dec 2009

Posts: 150
Thanks: 0

Re: determine pairs of functions independent

The following representation of may be of help to you:
Squaring x gives a positive number, even if x was negative. Then the square root acts on the positive result to give back a positive result. So whether or not x is negative, returns the positive magnitude of .

Using the chain rule we differentiate the function:


Thus the product rule gives us that:

I used the fact that to get to the second to last step.

So let's compute the Wronskian:

We can see that if and only if

AN ASIDE:
In the alternative definition of I gave above, it should be noted that I am saying that is the composition of the squaring function with the square root function . Reversing the order in which you compose the two functions (square rooting before squaring) would require restricting the domain of to , which by virtue of its smaller domain is not equal to the absolute value function. If you were to consider on the whole real line you would have to have the range of by the union of the axis of the complex plane (all pure imaginary and purely real numbers), and the result for negative x is not what we want either:
Let , then , where
forcesofodin is offline  
November 8th, 2010, 12:28 PM   #5
Math Team
 
Joined: Apr 2010

Posts: 1,958
Thanks: 114

Re: determine pairs of functions independent

Nice post, forcesofodin!
Quote:
Originally Posted by forcesofodin
[latex]g&#39;(x) \, = \, (2x)|x| + x^2 (\frac{x}{|x|}) \,
From here on, you can also use that
So that


Hoempa
Hoempa is offline  
November 9th, 2010, 05:18 AM   #6
 
Joined: Sep 2010

Posts: 73
Thanks: 0

Re: determine pairs of functions independent

Wow, that was a lovely post! Thanks for clearing it up
mbradar2 is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
determine, functions, independent, pairs



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Independent or Matched Pairs wyoung Algebra 3 April 27th, 2012 03:01 PM
Determine all pairs (x,y) TPB Algebra 1 November 13th, 2010 02:39 AM
Determine integer pairs NQrgArd Number Theory 1 November 26th, 2007 05:59 AM
Linearly independent linear functions mathiask Linear Algebra 1 October 23rd, 2007 12:13 AM
linear independent or dependent set of functions 1thoughtMaze1 Calculus 5 January 1st, 1970 12:00 AM





Copyright © 2014 My Math Forum. All rights reserved.