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baldymcgee6 Group Title

First and second derivative analysis... Is this right so far?

  • 2 years ago
  • 2 years ago

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  1. baldymcgee6 Group Title
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    http://screencast.com/t/5vrFs8Ggn

    • 2 years ago
  2. baldymcgee6 Group Title
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    @zepdrix can you help?

    • 2 years ago
  3. zepdrix Group Title
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    analysis? what does that mean? like you're trying to find critical points and such? :o

    • 2 years ago
  4. baldymcgee6 Group Title
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    Yeah, sorry... Max, min, inflections..

    • 2 years ago
  5. zepdrix Group Title
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    We have critical points anywhere the first derivative is 0 or UNDEFINED. They won't necessarily give us max or min values, but they are critical points, giving us some information :O \[f'(x)=\frac{ \sqrt{1+x} }{ \sqrt{1-x} }\cdot \frac{-1}{(1+x)^2}\]\[f'(x)=-\frac{1}{ \sqrt{1-x} \sqrt{1+x}}\]

    • 2 years ago
  6. zepdrix Group Title
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    Did i simplify that correctly? The first square root term you had, the -1/2 power, i just flipped the fraction so it wasn't negative anymore :O

    • 2 years ago
  7. baldymcgee6 Group Title
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    Didn't know you can do that.. :/

    • 2 years ago
  8. zepdrix Group Title
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    Hmm if you graph this on wolfram, you'll see that the critical points are actually telling us where the asymptotes are located :D http://www.wolframalpha.com/input/?i=y%3D%28%281-x%29%2F%281%2Bx%29%29%5E%281%2F2%29

    • 2 years ago
  9. zepdrix Group Title
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    Maybe not the most relevant if we just want max and min values :) but still kinda neat.

    • 2 years ago
  10. zepdrix Group Title
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    err asymptote at -1 at least.. 1 something else is going on maybe.. hmm i can't tell by that graph hehe

    • 2 years ago
  11. zepdrix Group Title
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    blah i dunno XD

    • 2 years ago
  12. baldymcgee6 Group Title
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    hmm, interesting. So we know a critical point is x=1 right?

    • 2 years ago
  13. zepdrix Group Title
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    yah it looks like x=1 and x=-1 are critical points. :D

    • 2 years ago
  14. zepdrix Group Title
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    If i did my math correctly.. :O i hope.

    • 2 years ago
  15. baldymcgee6 Group Title
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    yeah, x=1, -1... But what about domain issues?

    • 2 years ago
  16. zepdrix Group Title
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    Oh good call, -1 isn't in our domain :3 so we don't care about that value.

    • 2 years ago
  17. baldymcgee6 Group Title
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    hmmmmm... i'm a little bit confused... what IS our domain?

    • 2 years ago
  18. zepdrix Group Title
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    Hmmm so we can't divide by 0, so -1 is out. Anddd we can't take the square root of a negative number, meaning, Anything smaller than -1 is out (it will make the bottom negative). Anything larger than 1 is out (it will make the top negative). I think our domain looks like this. \[-1 < x \le 1\]

    • 2 years ago
  19. baldymcgee6 Group Title
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    That looks right, automatically the end points of our domain are critical values right?

    • 2 years ago
  20. baldymcgee6 Group Title
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    Because we can't attach a tangent line there?

    • 2 years ago
  21. zepdrix Group Title
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    Automatically? :o hmm i dunno. I just know that they're critical points because the first derivative told us so. It turns out that this is probably one of those weird functions where we'll end up having a MINIMUM at x=1 even though it doesn't have a slope of 0 there :D

    • 2 years ago
  22. zepdrix Group Title
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    Maybe that's what you were saying though :)

    • 2 years ago
  23. baldymcgee6 Group Title
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    Well, our prof said that the end points of the domain are critical values, because we cannot attach a tangent line there... I'm not sure I really understand that, I had never heard that before.

    • 2 years ago
  24. zepdrix Group Title
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    Hmm interesting :D

    • 2 years ago
  25. baldymcgee6 Group Title
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    I just found this: http://screencast.com/t/b4Jx1qLkQlo

    • 2 years ago
  26. zepdrix Group Title
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    cool :O

    • 2 years ago
  27. baldymcgee6 Group Title
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    I'm still unsure of our critical points though, if x=-1 is outside of our domain, then we can't use that,.. so I guess just x=1 is critical?

    • 2 years ago
  28. zepdrix Group Title
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    yah, f'(x)=0 will give us critical points (which there were none). and also a and b are considered critical points. But b wasn't in our domain, so we don't even consider that point. Yah sounds good :D So just the lonely little x=1.

    • 2 years ago
  29. baldymcgee6 Group Title
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    • 2 years ago
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  30. baldymcgee6 Group Title
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    I don't understand how there is a minimum at x=1.... We can't use a test value x>1 because it is out of our domain.. again :(

    • 2 years ago
  31. zepdrix Group Title
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    |dw:1353277622511:dw| True, we can't test both sides, all we can do is this.. see that it's decreasing on the left, and nonexistant on the right. So it's a minimum.

    • 2 years ago
  32. baldymcgee6 Group Title
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    oohhh.. okay... now to move onto the second derivative tests... :)

    • 2 years ago
  33. zepdrix Group Title
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    I would probably use the simplified form that I had set up, I think it'll be easier to differentiate again :)

    • 2 years ago
  34. zepdrix Group Title
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    \[\huge f'(x)=-(x-1)^{-1/2}(x+1)^{-1/2}\]

    • 2 years ago
  35. baldymcgee6 Group Title
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    right :D

    • 2 years ago
  36. zepdrix Group Title
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    Hmm this one is a pain in the butt, I'm coming up with: \[\huge f''(x)=\frac{x}{(x-1)^{3/2}(x+1)^{3/2}}\] It's quite possible that I made a mistake though :D So make sure you check your work.

    • 2 years ago
  37. zepdrix Group Title
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    I wanted to simplify it into a form that we can easily find inflection points from. So I combined the fractions :O

    • 2 years ago
  38. zepdrix Group Title
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    If you need to see my steps, I can post them :D

    • 2 years ago
  39. baldymcgee6 Group Title
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    Yep got that too!

    • 2 years ago
  40. baldymcgee6 Group Title
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    so... When f''(x) = 0 x = 0

    • 2 years ago
  41. zepdrix Group Title
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    If you refer back to our graph on wolfram, you'll see that makes sense! It changes from CCU to CCD around x=0!

    • 2 years ago
  42. baldymcgee6 Group Title
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    barely.. lol

    • 2 years ago
  43. zepdrix Group Title
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    XD

    • 2 years ago
  44. zepdrix Group Title
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    If you ever use Wolfram, make sure you're looking at the "Real-valued plot" from the graph drop down box.

    • 2 years ago
  45. baldymcgee6 Group Title
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    haha, yeah.. been there before.. I prefer the TI

    • 2 years ago
  46. zepdrix Group Title
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    oh fair enough c:

    • 2 years ago
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