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anonymous

  • one year ago

Calculus Help!

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  1. anonymous
    • one year ago
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    Rate of Change: Find the instantaneous rate of change of the surface area S = 6x^2 of a cube with respect to the edge length x at x = a.

  2. DanJS
    • one year ago
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    instantaneous rate of change, also known as the derivative in this case it will be with respect to side length x

  3. DanJS
    • one year ago
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    need to take \[\frac{ d }{ dx }\] of both sides

  4. anonymous
    • one year ago
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    Wait what is d/dx?

  5. DanJS
    • one year ago
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    the derivative with respect to x, the side length, or sometimes it could be maybe, d/dt , with respect to time, or anything else

  6. DanJS
    • one year ago
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    wait, what have you learned so far? just limits?

  7. anonymous
    • one year ago
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    Ya I only learned that.

  8. DanJS
    • one year ago
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    so you have to use the limit definition of the instantaneous rate of change then i guess

  9. anonymous
    • one year ago
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    How would you do that?

  10. DanJS
    • one year ago
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    the [ f(x)+delta x) - f(x)] / delta x

  11. anonymous
    • one year ago
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    Isn't that \[\frac{ f(x+h)-f(x) }{ h }\]?

  12. anonymous
    • one year ago
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    Is that the same thing?

  13. DanJS
    • one year ago
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    \[\lim_{\Delta x \rightarrow 0}\frac{ f(x+\Delta x )-f(x)}{ \Delta x }\]

  14. DanJS
    • one year ago
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    yeah

  15. anonymous
    • one year ago
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    Okay so what would be h and what would be x?

  16. DanJS
    • one year ago
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    Trust me , the class will get easier after this , well less work at least

  17. anonymous
    • one year ago
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    Does f(x)=6x^2?

  18. anonymous
    • one year ago
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    Ya I wish I had a good teacher though. She doesn't even teach!!

  19. DanJS
    • one year ago
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    use khan academy, lots of vids there, or MIT opencourse ware... even better if you have time

  20. anonymous
    • one year ago
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    She can't "explain the method". She can solve it but she can't explain it. I don't see why she is teaching.

  21. anonymous
    • one year ago
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    Ya I was able to understand most of it. I just couldn't find the ones talking about Rate of Change.

  22. DanJS
    • one year ago
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    rate of change is just like from algebra rise over run -- a slope

  23. DanJS
    • one year ago
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    y has a rate of change for each x change

  24. anonymous
    • one year ago
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    Sorry but that was just my opinion about her.

  25. anonymous
    • one year ago
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    So plug in x in 6x^2?

  26. anonymous
    • one year ago
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    x=0

  27. anonymous
    • one year ago
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    So y=0?

  28. DanJS
    • one year ago
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    but the derivative you are gonna learn will show you what the slope is of a tangent line to a curve

  29. DanJS
    • one year ago
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    there is only 1 point how can that have a slope?

  30. DanJS
    • one year ago
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    that is where the limit we are doing here comes from... as the change in x shrinks infinitley small, then you can have an 'instantaneous' rate of change...the rate of change of a single point? sounds messed up , but that is the power of calc

  31. DanJS
    • one year ago
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    that is all the derivative is, then you will learn like 10 shortcuts and ways to find derivatives for like 2 months

  32. DanJS
    • one year ago
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    sorry i got off track

  33. anonymous
    • one year ago
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    It's fine. What would I do with a if x=a?

  34. DanJS
    • one year ago
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    Rate of Change: Find the instantaneous rate of change of the surface area S = 6x^2 of a cube with respect to the edge length x at x = a.

  35. DanJS
    • one year ago
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    the function is S(x) surface area as a function of length of side

  36. anonymous
    • one year ago
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    Okay

  37. DanJS
    • one year ago
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    so this is a dynamic system of a box growing or shrinking, they want to know the rate the surface area is changing the moment the side length x is a (x=a)

  38. DanJS
    • one year ago
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    so in the limit definiton use S(a) = 6a^2 and plug all that in

  39. anonymous
    • one year ago
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    But then what would a equal?

  40. DanJS
    • one year ago
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    a is just a constant arbitrary value

  41. anonymous
    • one year ago
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    S(a) =6a^2 is what then?

  42. DanJS
    • one year ago
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    so that is the surface area value when x=a...

  43. anonymous
    • one year ago
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    Is that it?

  44. DanJS
    • one year ago
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    plug s(a) in for the f(x) in that limit definition of inst. rate of change

  45. anonymous
    • one year ago
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    Where would S(a) be plugged into? f(x+h) or f(x) or h?

  46. DanJS
    • one year ago
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    \[\lim_{h \rightarrow 0}\frac{ S(x+h) - S(x) }{ h }=\]

  47. DanJS
    • one year ago
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    actually just keep it as the general S(x) function for now

  48. anonymous
    • one year ago
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    Okay so what is s(x+h) equal?

  49. DanJS
    • one year ago
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    s(x) = 6x^2 replace all x with (x+h) s(x+h) = 6(x+h)^2

  50. DanJS
    • one year ago
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    so you get \[\lim_{h \rightarrow 0}\frac{ 6(x+h)^2 - 6x^2 }{ h }\]

  51. DanJS
    • one year ago
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    you get it all so far?

  52. anonymous
    • one year ago
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    Ya I got that part.

  53. DanJS
    • one year ago
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    so you have to do the algebra now and see if you can take the limit... as h goes to zero... the denominator goes to zero... not good

  54. DanJS
    • one year ago
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    expand and combine the top part

  55. anonymous
    • one year ago
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    Okay hold on.

  56. anonymous
    • one year ago
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    12x+6h

  57. DanJS
    • one year ago
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    \[\lim_{h \rightarrow 0}\frac{ 6(x+h)^2 - 6x^2 }{ h }~~=~~\lim_{h \rightarrow 0}\frac{ 12hx+6h^2}{ h }\]

  58. DanJS
    • one year ago
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    --- (x+h)^2 = x^2 + 2xh + h^2

  59. DanJS
    • one year ago
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    you see the x^2 term canceled out nicely, 6x^2 - 6x^2

  60. DanJS
    • one year ago
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    got to hurry a bit,

  61. anonymous
    • one year ago
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    Ya I got that.

  62. DanJS
    • one year ago
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    If i remember, that is usually the case, the squared term cancels, so you can now factor an h out that will cancel with the h in the denominator...

  63. anonymous
    • one year ago
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    So the answer is -12x

  64. DanJS
    • one year ago
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    \[\lim_{h \rightarrow 0}(12x + 6h)\]

  65. DanJS
    • one year ago
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    positive 12 yeah, put h=0 in

  66. DanJS
    • one year ago
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    we can call that function the instantaneous rate of change of the surface area with respect to the side length x, for any length x.... name it S prime (x) ... S ' (x) = 12x

  67. anonymous
    • one year ago
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    So that's it?

  68. DanJS
    • one year ago
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    pretty much... So overall this is a box that is changing size at a constant linear rate (since 12x is a linear function)

  69. DanJS
    • one year ago
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    and now all they want you to do is calculate it when x=a S(x) = 6x^2 S ' (x) = 12x ---> S ' (a) = 12*a

  70. anonymous
    • one year ago
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    Okay thanks!

  71. DanJS
    • one year ago
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    let me show you something real fast that will help

  72. DanJS
    • one year ago
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    we can start calling the 'instantaneous rate of change now' the derivative... notice... S(x) = 6x^2 S ' (x) = 12x

  73. DanJS
    • one year ago
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    The first shortcut to finding the derivative is the simple power rule... for exponents like S(x) the rule is simply....

  74. DanJS
    • one year ago
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    the derivative of any term like polynomial terms \[\huge a*x^n\] is found by this simple formula...multiply by the exponent, then decrease that exponent by 1

  75. anonymous
    • one year ago
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    Okay. So how would the derivitive help in this?

  76. DanJS
    • one year ago
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    \[\huge n*a*x ^{n-1}\]

  77. DanJS
    • one year ago
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    the surface area function was S(x) = 6x^2 after all that work, we figured the derivative was S ' (x) = 12x

  78. DanJS
    • one year ago
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    all you have to do is... follow that power formula... 6*2*x^(2-1)

  79. anonymous
    • one year ago
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    Wow!!! That is seriously so simple with that method!!

  80. DanJS
    • one year ago
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    or 12x

  81. DanJS
    • one year ago
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    haha, yeah, but the limit stuff helps to understand better what the derivative is...

  82. DanJS
    • one year ago
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    so the derivative of f(x) = 10x^3 + 12x^2 + 3

  83. DanJS
    • one year ago
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    f '(x) = 30x^2 + 24x

  84. DanJS
    • one year ago
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    derivative of a constant number is zero

  85. DanJS
    • one year ago
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    the limit thing we did will get the same answer with all the algebra...

  86. DanJS
    • one year ago
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    aight, got to go.. goodluck

  87. anonymous
    • one year ago
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    Thanks!

  88. DanJS
    • one year ago
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    ill add you as friend and can help ya out whenever i am on

  89. anonymous
    • one year ago
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    That would be great! I'll do the same. :)

  90. DanJS
    • one year ago
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    no prob, i would rather recall and relearn calc again than do another system of linear equations .. hah bye

  91. anonymous
    • one year ago
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    Alright bye!

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