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Babynini

  • one year ago

greatest integer functions. Help!

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  1. Babynini
    • one year ago
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    #51

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  2. Babynini
    • one year ago
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    @SolomonZelman you free? :)

  3. jim_thompson5910
    • one year ago
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    do you know how the greatest integer function is defined?

  4. Babynini
    • one year ago
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    Mm I know what the graph of it looks like. But not quite sure how it's defined

  5. jim_thompson5910
    • one year ago
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    http://mathbits.com/MathBits/TISection/PreCalculus/GraphGreatestIntFunction.html the basic idea is that the input x could be any real number the output is always a whole number (positive or negative). You round down to the nearest whole number

  6. jim_thompson5910
    • one year ago
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    greatest integer function = floor function

  7. jim_thompson5910
    • one year ago
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    think of the number line written vertically |dw:1444263692393:dw|

  8. jim_thompson5910
    • one year ago
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    if we plug in x = 1.7 then [[x]] = [[1.7]] = 1 we round down to the nearest whole number which in this case is 1 |dw:1444263763896:dw|

  9. jim_thompson5910
    • one year ago
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    now let's say we plugged in x = -0.16 that would move to -1 because we're moving down the ladder or building |dw:1444263816583:dw|

  10. Babynini
    • one year ago
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    approaching it from either the positive or negative side we always do that?

  11. SolomonZelman
    • one year ago
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    Here are the properties, which follow just from logic: *[1]* \(\Large\color{black}{ \displaystyle \lim_{x~\to~a^+} \left[\left[x\right]\right]=a }\) *[2]* \(\Large\color{black}{ \displaystyle \lim_{x~\to~a^-} \left[\left[x\right]\right]=a-1 }\) where *a* is an integer.

  12. jim_thompson5910
    • one year ago
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    let's try an example if f(x) = [[x]], then what is f(8.3125) equal to?

  13. Babynini
    • one year ago
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    8?

  14. jim_thompson5910
    • one year ago
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    yes, how about f(8.99999)

  15. Babynini
    • one year ago
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    8 still

  16. jim_thompson5910
    • one year ago
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    good, so if the number is positive, you just chop off the decimal portion

  17. jim_thompson5910
    • one year ago
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    now let's try f(-2.462)

  18. Babynini
    • one year ago
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    Would that round up? to 3?

  19. jim_thompson5910
    • one year ago
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    I think you meant -3

  20. jim_thompson5910
    • one year ago
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    look back to the vertical number line

  21. Babynini
    • one year ago
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    yes yes sorry o.o

  22. jim_thompson5910
    • one year ago
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    |dw:1444264095476:dw|

  23. jim_thompson5910
    • one year ago
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    think of each whole number as a floor in a building -2.462 is between the two floors the floor function moves -2.462 down to the nearest floor

  24. Babynini
    • one year ago
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    so approaching -2 from the right = -2 and from the left = -2 ?

  25. Babynini
    • one year ago
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    because there's no decimals o.o

  26. jim_thompson5910
    • one year ago
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    look at the rules SolomonZelman posted

  27. Babynini
    • one year ago
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    Ahh ok. Approaching - 2 from the right = 2 approaching - 2 from the left = ...1?

  28. jim_thompson5910
    • one year ago
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    idk how you jumped from -2 to +2

  29. Babynini
    • one year ago
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    *-2 and -3

  30. jim_thompson5910
    • one year ago
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    \[\Large \lim_{x \to -2^{+}} [[x]] = -2\] \[\Large \lim_{x \to -2^{-}} [[x]] = -3\] looks good

  31. Babynini
    • one year ago
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    Sorry, my brains are not working haha =.=

  32. jim_thompson5910
    • one year ago
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    does \[\Large \lim_{x \to -2^{}} [[x]]\] exist?

  33. Babynini
    • one year ago
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    and then as x approaches -2.4 it = -3

  34. Babynini
    • one year ago
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    No, it doesn't.

  35. jim_thompson5910
    • one year ago
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    why not?

  36. Babynini
    • one year ago
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    because the points aren't meeting

  37. jim_thompson5910
    • one year ago
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    yes, specifically because \(\Large \text{LHL} \ne \text{RHL}\) LHL = left hand limit RHL = right hand limit

  38. jim_thompson5910
    • one year ago
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    so 51(a)(ii) does not exist

  39. Babynini
    • one year ago
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    ahh gotcha. Okay!

  40. Babynini
    • one year ago
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    Part b!

  41. Babynini
    • one year ago
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    for c) the answer is: for all non-integer values of a, yeah?

  42. jim_thompson5910
    • one year ago
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    `for c) the answer is: for all non-integer values of a, yeah?` agreed

  43. jim_thompson5910
    • one year ago
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    part b) is very similar to what SolomonZelman wrote out

  44. Babynini
    • one year ago
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    Okay, I wasn't sure if that was too "simple" So b: i) n-1 ii) n

  45. jim_thompson5910
    • one year ago
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    looks good

  46. Babynini
    • one year ago
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    Thank you so much :)

  47. jim_thompson5910
    • one year ago
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    sure thing

  48. Empty
    • one year ago
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    Oh I thought this was going to be another squeeze theorem thing, but in case you come across one soon, I think you might be entertained: \[x \le [[x]] \le x+1\] So if you're given like find the limit of \(x [[x]]\) as x approaches 0 you already know: \[x^2 \le x[[x]] \le x^2+x\] So \[\lim_{x \to 0} x[[x]] = 0\] Anyways there are a lot of problems with squeeze theorem and this rounding function in it so I thought it'd be fun to share real fast for fun.

  49. Babynini
    • one year ago
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    hahah my next problem is that! i'll tag you in it ;)

  50. Babynini
    • one year ago
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    Though from all you said here I think I could get it.

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