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Complicated Calculus question

  1. Jul 27, 2011 #1
    1. The problem statement, all variables and given/known data
    Microbiologists contribute their expertise to many fields, including medicine,environmental science, and biotechnology. Enumerating, the process of countingbacteria, allows microbiologists to build mathematical models that predict populations after a given amount of time has elapsed. Once they can predicta population accurately, the model can be used in medicine, for example, topredict the dose of medication required to kill a certain bacterial infection. The data in the table shown was used by a microbiologist to produce a polynomial-based mathematical model to predict population p (t) as a function of time t, in hours, for the growth of acertain strain of bacteria:

    p (t) = 1000( 1 + t + 1/2 t^2 + 1/6 t^3 + 1/24 t^4 + 1/120 t^5 )

    time (h)- population
    0.0-- 1000
    0.5-- 1649
    1.0-- 2718
    1.5-- 4482
    2.0-- 7389

    How well does the function fit the data? Use the data, the equation, a graph, and/or a graphing calculator to comment on the “goodness of fit.”

    Use p(t) and p'(t) to determine the following:
    a)the population after 0.5 h and the rate at which the population is growing at this time.
    b)the population after 1.0 h and the rate at which the populationis growing at this time.

    What pattern did you notice in your calculations? Explain this pattern by examining the terms of the equation to find the reason why


    2. Relevant equations
    p (t) = 1000( 1 + t + 1/2 t^2 + 1/6 t^3 + 1/24 t^4 + 1/120 t^5 )


    3. The attempt at a solution
    i subbed each time value in the p(t) equation and i got the population (it matches the given table) i don't know how to explain this. i got the derivative of p(t) but i'm not sure if it is right. can someone post the derivative of p (t).
     
  2. jcsd
  3. Jul 27, 2011 #2

    Dick

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    What did YOU get for p'(t)? Then we'll tell you if it's right.
     
  4. Jul 27, 2011 #3
    p'(t) = 1000( 1 + t + 1/2 t^2 + 1/6 t^3 + 1/24 t^4 )
     
  5. Jul 27, 2011 #4

    Dick

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    Right!
     
  6. Jul 27, 2011 #5
    i got:
    p(0.5) = 1648.69 and p'(0.5)= 1648.43166
    p(1) = 2716.7 and p'(1) = 2708.367 is this correct?

    i don't know how to explain the pattern
     
  7. Jul 27, 2011 #6

    Dick

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    Yes, those are approximately correct. I think you are doing the numbers correctly. On the other hand, it's maybe a little difficult to figure out what you are expected to observe. Just tell me off of the top of your head, what do you observe?
     
  8. Jul 27, 2011 #7
    as time increases, the difference between p(t) and p'(t) also increases.
    this is what i observe. i have no explanation to why?
     
  9. Jul 27, 2011 #8

    Dick

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    That's a start. I would also observe that p(t) and p'(t) are approximately equal at all times, but as you say, getting worse at late times. I would also observe that p(2)=7266.67 is also not a terribly good approximation to the given data point 2--7389. So the polynomial approximation is not fitting the data very well for large values of t. There might be a better approximation form than the polynomial. I'm taking a wild guess they might want you to guess what function form this might be.
     
  10. Jul 27, 2011 #9
    what do you mean by function form???
     
  11. Jul 27, 2011 #10

    Dick

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    I mean the microbiologist is using a polynomial to model the bacteria growth curve. There are other functions of t besides polynomials. Do you know some? I do. log(t), sin(t) etc etc. Those aren't polynomials. Can you think of one that might be relevant if you observe p(t) is approximately equal to p'(t)??
     
  12. Jul 27, 2011 #11
    ughhhhh! lol i'm not sure unmmmm linear?
     
  13. Jul 27, 2011 #12

    Dick

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    Linear is a form of polynomial. You are taking calculus, right? There are more. What function is equal to its derivative?? If it's late and you are tired, that's ok. Try it in the morning. Actually the question might not be going that far. Can you see why p(t) is close to p'(t) from your equations for both? This is where I get unsure exactly what you are SUPPOSED to conclude.
     
    Last edited: Jul 27, 2011
  14. Jul 28, 2011 #13

    Ray Vickson

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    Plot the data. Does it look like a straight line to you?

    RGV
     
  15. Jul 29, 2011 #14
    Hint:

    Many functions modeling bacterial growth, population growth, decay, use natural logs and the inverse function of natural logs (about as close as a hint as I can give without naming it). Thinking around these lines, what function involving the aforementioned general family has a derivative which is the same as itself assuming you have just x involved?
     
  16. Jul 29, 2011 #15
    Hint:

    Many functions modeling bacterial growth, population growth, decay, use natural logs and the inverse function of natural logs, (about as close as a hint as I can give without naming it), and exponentials. Thinking around these lines, what function involving the aforementioned general family has a derivative which is the same as itself assuming you have just x involved?
     
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