|Apr30-07, 06:54 AM||#1|
1. The problem statement, all variables and given/known data
If f(x)=| log x | , then state whether the following are true or false :
c)y'(0)= ∞ (infinity)
2. Relevant equations
|y| = y if y >=0
|y| = -y if y<0
3. The attempt at a solution
a)The first one is true because if x > 1, then y > 0 and | y | is y. So, y' = 1/x
b)If x=1, y=0 and so |y| = y . So y' =1/x = 1
c)If x=0, y is not defined ?
I am having trouble with b) and c) . I dont understand whether to check by putting values first (which would give all constant, and IMO is wrong) or later as I have done.
The answers are T,F,F
Any help is appreciated
|Apr30-07, 07:41 AM||#2|
Blog Entries: 1
For part b, (I'm assuming y=f), y'(x) exists if y'(x+) = y'(x-) (this is for a particular x). So you need to show the derivative from the left = the derivative from the right (or show it doesn't)
Otherwise, for example, let y=|x|. y(0) = 0, so y'(0) = x' = 1. But this is clearly false.
For part c, I'm tempted to guess they want y'(x) as x->0 from above (otherwise you're absolutely correct). This is actually negative infinity (it cna be seen just by graphing y), so it's false either way.
|Apr30-07, 12:24 PM||#3|
"y'(1+)= 1/x" What does "1+" mean? OfficeShreder interpreted it to mean the "right derivative" or "limit from the right" but if that were correct and you are asking for the derivative as you approach x= 1 from the right, then there would be no "x" in the derivative.
If x> 1 then f(x)= log x. What is the derivative of that? What is the limit of the derivative as x-> 1 from above? If x< 1, then f(x)= - log(x). What is the derivative of that? What is the limit of that as x-> -1 from below?
Yes, you are correct that f(0) is not defined and neither is f'(0). I personally don't like saying that something is infinity when it is not defined! However, here, I think you need to look at what happens to |log x| and its derivative for numbers like x= 0.0001, x= 0.00000001, etc.
|May1-07, 05:40 AM||#4|
Thx OfficeShredder and Halls for the help.
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