Some calculus help?

[Soccer55]

Originally posted by: BrownTown
derivitive of |8x - 56| = 8
derivitive if (7 - x) = -1

8/-1 = -8

That seems right, not sure if the ABS is gonna mess it though

Yes, the absolute value messes it up. The derivative of |x| is sgn(x), not 1.

-Tom

 

[BrownTown]

Did I mention that limits are stupid...

 

[hypn0tik]

Break it up into two cases since you have an absolute value:

Case 1:
8x-56 > 0 ==> |8x-56| = 8x-56 ==> f(x) = (8x-56)/(7-x)=8(x-7)/(7-x) = -8
But, 8x-56 > 0 means that 8x > 56 and x > 7.


Case 2:
8x-56 < 0 ==> |8x-56|=56-8x ==> f(x) = (56-8x)/(7-x) = 8(7-x)/(7-x) = 8
But, 8x-56 < 0 means that 8x < 56 and x < 7

In the question, you are told that x approaches 7 from below, therefore, x < 7 and we are interested in Case 2.

 

[SleepWalkerX]

Originally posted by: Soccer55

Originally posted by: SleepWalkerX

Originally posted by: Soccer55
The absolute value function is not a step function, so you wouldn't want to look at it like that. You're probably thinking of a piecewise function, not step function. So to do what you want, you would have to figure out the equations of the lines that would make up the piecewise function and make sure you specify the domain where each part is valid. For example: f(x) = | x | can be written as a piecewise function with 2 parts. f(x) = x for x >= 0 and -x for x < 0.

-Tom

Oops, you're right I meant piecewise. I'm still kinda confused. Should I be worrying about setting up the function as a piecewise function at all?

Well, here's what I'd do. We know that |a*b| = |a|*|b|, so |8x-56| = |-56+8x| = |-8*(7-x)| = |-8|*|7-x| = 8*|7-x|. The reason I do that is since we're looking at this limit from the left, 7-x will always be positive. Therefore, |7-x| = 7-x. So now when you take the limit from the left, you have to look at 8*(7-x)/(7-x) = 8.

Note: |7-x| = 7-x is only true because we're looking at the limit from the left as x -> 7. If we were looking at the limit from the right, it would not be true.

-Tom

Oh ok this makes sense.

Thanks for everyone who replied in this thread. I greatly appreciate it. 😀

 

[habib89]