My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
December 12th, 2015, 02:19 PM   #1
Member
 
Joined: May 2015
From: Earth

Posts: 64
Thanks: 0

Limit rules trick question?

Use the definition of the derivative (and limit rules) to find the derivative of |x^3| at x = 0

Is this a trick question? Like just from looking at it, I know the derivative of an absolute value such as x^3 does not equate at x=0, right?
eglaud is offline  
 
December 12th, 2015, 02:58 PM   #2
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 2,922
Thanks: 1518

$\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

$\displaystyle f'(0) = \lim_{x \to 0} \frac{|x^3| - 0}{x - 0}$

$x < 0 \implies |x^3| = -x^3$

$\displaystyle \lim_{x \to 0^-} \frac{-x^3}{x}$

$\displaystyle \lim_{x \to 0^-} -x^2 = 0$

$x \ge 0 \implies |x^3| = x^3$

$\displaystyle \lim_{x \to 0^+} \frac{x^3}{x}$

$\displaystyle \lim_{x \to 0^+} x^2 = 0$

limit from the left = limit from the right, therefore ...

$\displaystyle f'(0) = \lim_{x \to 0} \frac{|x^3| - 0}{x - 0} = 0$
skeeter is online now  
December 12th, 2015, 04:53 PM   #3
Member
 
Joined: May 2015
From: Earth

Posts: 64
Thanks: 0

So you're saying apply the sandwich rule or whatever? To take the derivative on both sides to get the limit?

Upon looking at the graph I see there is no edge there like I thought there was. If there was an edge, there would be no limit there, or derivative? How does that all I work? I forget. Thanks.
eglaud is offline  
December 12th, 2015, 06:23 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 20,619
Thanks: 2074

It's straightforward to show from first principles that ax³ has derivative 3ax², with value 3a*0 = 0 at x = 0. Letting a = sgn(x) doesn't affect this result.

That wouldn't work for |x|, as the derivatives of x and -x take different values at x = 0.

The derivative of |x³| is 3x|x|, as can be verified by differentiating (x²)$^{3/2}$ using the usual rules.
skipjack is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
limit, question, rules, trick



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Simple or trick question? MetallicaBand Elementary Math 3 June 10th, 2014 10:54 AM
Is this a trick question? Converting Shamieh Algebra 3 May 29th, 2013 06:39 AM
trick question mathkid Algebra 3 February 25th, 2013 03:51 AM
Is this Trick Question?? mathkid Calculus 1 September 22nd, 2012 11:42 AM
A question on limit rules tomorrow Calculus 14 September 22nd, 2012 03:05 AM





Copyright © 2019 My Math Forum. All rights reserved.