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RobertSn

  • 2 years ago

Hey guys, I am looking for some help with calculus. It is in regard to rieman sums, and I would really appreciate any help understanding. I will post links below and explain further.

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  1. RobertSn
    • 2 years ago
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    The graph shown is y=x I just dont see how those steps were derived

  2. RobertSn
    • 2 years ago
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    And this is part of it also

  3. RobertSn
    • 2 years ago
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    Is therom 5 something that must be memorized? Or is there any logic to it?

  4. kc_kennylau
    • 2 years ago
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    |dw:1388473083421:dw|

  5. RobertSn
    • 2 years ago
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    What does k represent?

  6. kc_kennylau
    • 2 years ago
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    k is from 1 to n, if you notice the summation sign in your first photo

  7. RobertSn
    • 2 years ago
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    But why is that over n the height?

  8. RobertSn
    • 2 years ago
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    And is the width 1/n just because we are taking it as n approaches infinity? so therefor 1/n is approach infinitely small?

  9. RobertSn
    • 2 years ago
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    @kc_kennylau ?

  10. kc_kennylau
    • 2 years ago
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    yes, and sorry I'm not too familiarized in this topic... furthermore i'm busy :'(

  11. kc_kennylau
    • 2 years ago
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    sorry :'(

  12. RobertSn
    • 2 years ago
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    Thanks for the input anyways kc!

  13. myininaya
    • 2 years ago
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    Ok I can try drawing a picture for you and posting it. But first let's see if you can see the pattern first. so the very first number is a_1=(1-1)/n the next number is a_2=(2-1)/n=1/n <---this is how for we are away from 0 or a_1 since a_1 is 0 a_3=(3-1)/n .... now going to the kth interval where we have the x-value being a_k=(k-1)/n ... going all the way to the nth interval we have a_n=(n-1)/n because we are plugging in the left end point of that interval and not the right endpoint because we are doing left-endpoint rule we are taking all the left endpoints of all the n intervals beginning at 0

  14. myininaya
    • 2 years ago
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    Now to find the heights of the rectangles we do f(a_k)

  15. RobertSn
    • 2 years ago
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    By a_1 does that mean a/1?

  16. myininaya
    • 2 years ago
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    |dw:1388474479972:dw|

  17. myininaya
    • 2 years ago
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    a_1 usually means a subscript 1

  18. myininaya
    • 2 years ago
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    the base of each rectangle is 1/n

  19. RobertSn
    • 2 years ago
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    Alright, okay so f(k_n) would be f(k-1/n)

  20. RobertSn
    • 2 years ago
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    and thats the height at any given k

  21. myininaya
    • 2 years ago
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    you mean f(a_k)?

  22. myininaya
    • 2 years ago
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    If so yes

  23. myininaya
    • 2 years ago
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    but since this function is f(x)=x then f((k-1)/n)=(k-1)/n

  24. RobertSn
    • 2 years ago
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    Alright I think it is starting to make sense, but when I try an example I still just dont exactly see it. Ill post if you could take a look,

  25. myininaya
    • 2 years ago
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    Ok, so looking at this example, can you tell me what puzzles you and I will see if I can explain it?

  26. RobertSn
    • 2 years ago
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    Sure just a moment

  27. RobertSn
    • 2 years ago
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    Well first of all, do we know that it is left hand because of the n-1 on top of the sigma?

  28. RobertSn
    • 2 years ago
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    If that is so, for left handed questions like this can we always do (1/n)f(k/n)?

  29. RobertSn
    • 2 years ago
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    So we just set that equal to the function that is given?

  30. myininaya
    • 2 years ago
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    yep they started at 0 and when to n-1 so left endpoint

  31. RobertSn
    • 2 years ago
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    And what does the k=xn really represent?

  32. myininaya
    • 2 years ago
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    We are trying to find what f(x) is

  33. myininaya
    • 2 years ago
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    \[\int\limits_{a}^{b}f(x) dx=\sum_{i=0}^{n-1} \Delta x f(a+i \cdot \Delta x)\]

  34. myininaya
    • 2 years ago
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    where delta x =(b-a)/n

  35. RobertSn
    • 2 years ago
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    Right, and Xi= a+ i deltax/n

  36. myininaya
    • 2 years ago
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    the easiest thing to do is assume a equals 0 so we can just look at i*delta x

  37. RobertSn
    • 2 years ago
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    Oh i see, and how about the 1?

  38. myininaya
    • 2 years ago
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    well they chose the base to be 1/n which is delta x

  39. myininaya
    • 2 years ago
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    if we have (b-0)/n=1/n then b has to be?

  40. RobertSn
    • 2 years ago
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    1

  41. myininaya
    • 2 years ago
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    now this is one way to do the problem you don't have to choose it this way and the integral will still have the same value and yes b=1 for this example

  42. myininaya
    • 2 years ago
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    we can take this same problem and go a different way though

  43. myininaya
    • 2 years ago
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    would you like to?

  44. RobertSn
    • 2 years ago
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    Yes sure!

  45. myininaya
    • 2 years ago
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    I still want to choose a to be 0 because yeah that is just plain easiest!

  46. myininaya
    • 2 years ago
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    but with if we choose the intervals to have width 2/n instead of 1/n

  47. myininaya
    • 2 years ago
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    then b would have to be what?

  48. RobertSn
    • 2 years ago
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    then b would be two

  49. RobertSn
    • 2 years ago
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    So are you also saying that you can choose some of the conventions, as long as they match up?

  50. myininaya
    • 2 years ago
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    yes so but now we have \[\frac{2}{n} f(2 \cdot \frac{k}{n}) \text{ instead of } \frac{1}{n} f(\frac{k}{n}) \]

  51. myininaya
    • 2 years ago
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    yep

  52. myininaya
    • 2 years ago
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    like are answer will look different but it will still hold the same value

  53. myininaya
    • 2 years ago
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    our*

  54. myininaya
    • 2 years ago
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    \[\text{ so we want } \frac{2}{n} f(\frac{2k}{n})=\frac{1}{k+5n} \arctan(\frac{k+2n}{k+n})\]

  55. myininaya
    • 2 years ago
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    Solve for f(2k/n) by multiplying both sides by n/2

  56. RobertSn
    • 2 years ago
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    son/2 (k +5n)?

  57. myininaya
    • 2 years ago
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    \[f(\frac{2k }{n})=\frac{n}{2k+10n} \arctan(\frac{k+2n}{k+n})\]

  58. RobertSn
    • 2 years ago
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    Okay yea , and then how do you deal with the 2k/n?

  59. myininaya
    • 2 years ago
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    Now we want to know what f(x) is right?

  60. RobertSn
    • 2 years ago
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    right

  61. myininaya
    • 2 years ago
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    so what is k if 2k/n=x

  62. myininaya
    • 2 years ago
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    basically solve that for k so we can figure out what to replace k with so that we will just have x inside and not that other crap

  63. RobertSn
    • 2 years ago
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    Okay so we bring in an X ?

  64. RobertSn
    • 2 years ago
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    xn/2=k

  65. myininaya
    • 2 years ago
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    because we are trying to find out what integral notation looks like for this summation notation

  66. myininaya
    • 2 years ago
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    looks good so we will replace all the k's we see with that

  67. myininaya
    • 2 years ago
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    \[f(x)=\frac{n}{xn+10n}\arctan(\frac{\frac{xn}{2}+2n}{\frac{xn}{2}+n})\] \[f(x)=\frac{n(1)}{n(x+10)}\arctan(\frac{xn+4n}{xn+2n})=\frac{1}{x+10}\arctan(\frac{n(x+4)}{n(x+2)})\] \[f(x)=\frac{1}{x+10}\arctan(\frac{x+4}{x+2})\] so this is what our f(x) looks like from choosing a= 0 and b=2 there isn't a unique answer to your question the answer can totally vary

  68. myininaya
    • 2 years ago
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    basically you get to choose a and b

  69. myininaya
    • 2 years ago
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    and nothing else

  70. RobertSn
    • 2 years ago
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    Wow great. Thanks a lot for your help! Do you know the exact names of these types of problems? Im looking to find some practice problems

  71. myininaya
    • 2 years ago
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    This is just some applications to the definition of reimann sums let me see if i can find you some problems one moment

  72. myininaya
    • 2 years ago
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    http://freedom.mysdhc.org/teacher/1541derflingerk/documents/Calculus/FOV1-001B4535/Integration%20via%20Sigma(2).pdf these don't look at hard but you can try these 4.3 9-12

  73. myininaya
    • 2 years ago
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    as*

  74. myininaya
    • 2 years ago
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    they choose a and b for you

  75. RobertSn
    • 2 years ago
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    Alright, thanks!

  76. myininaya
    • 2 years ago
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    there c_i represents a+i*delta x by the way

  77. myininaya
    • 2 years ago
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    If you get bored of those attempt this one I made up: Write this as an integral: \[\lim_{n \rightarrow \infty} \sum_{i=0}^{n-1} \frac{i}{i+8n} \cos(\frac{i+3n}{n})\]

  78. eliassaab
    • 2 years ago
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    Try practicing on http://www.saab.org/calculus.cgi under Select What Kind Of Problems select Calculus I( Integrals(Substitution, FTC)

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