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Homework Help: Interpretation of differentiation results

  1. Mar 23, 2015 #1
    1. The problem statement, all variables and given/known data
    Please see the attached file. I am trying to understand the sensitivity of two related variables - Y and K - to an independent variable M.

    a. Is my differentiation of equation 2 correct?
    b. I can see that, based on eq. 4, K is more senstive to M than Y is, however I am not sure if I can quantify the difference. Would I be able to do that using the actual values of the constants - N, D, i, and G?
    c. How should I describe the dependency of K on M?

    2. Relevant equations
    Equations 1 and 2 in the attached file define Y and K. Equations 3 and 4 are derivatives of eqn. 1 and 2, respectively.

    3. The attempt at a solution
    The attached file shows my work.

    Attached Files:

    Last edited: Mar 23, 2015
  2. jcsd
  3. Mar 23, 2015 #2
    part a looks ok to me. not sure what you mean by sensitivity here. is it something to do with the size of the derivative?
  4. Mar 23, 2015 #3
    Thanks. Yes, by sensitivity I meant how quickly K is changing with a unit change in M.
  5. Mar 23, 2015 #4


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    For dependency, I think it is also valuable to look at asymptotics. Test ##M\to 0+, M\to 0-, M\to +\infty, M\to -\infty## and maybe some other values that make sense, maybe ##M = -D##.
  6. Mar 23, 2015 #5


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    Notice that you can write K as:
    ##\displaystyle k=y\cdot\frac{M+D}{M}-\frac{D\cdot i}{M}##​

    Then, substitute y into that.
  7. Mar 27, 2015 #6
    All variables are non negative. Here are some examples, N=5, D =50, y=7%, i = 4%, G =0.5.

    1. I wish to find out, at any particular value of M, say 60, whether y is more sensitive to a unit change in M than k is. How should I do that?

    2. I can visualize how function 3 will look but that is not true about to function 4. What kind of function is it? It has four M values in the denominator, two of which are squared. How should I think about such a complicated equation?
  8. Mar 27, 2015 #7


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    if the derivative is larger, it is more sensitive.
    Note that dy/dM = -N/(D+M)^2, so if M is 60, you have -N/(D^2+M*) where M*>3600. This is small. If M is 1, the you have -N(D^2+2D+1), so D is much more important.
    Use these same principles when looking at dk/dM.
    Usually, you can fix all the variables in the equation and only vary one at a time to see what the function does.
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