1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    RUber

    User Avatar
    Homework Helper

    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

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Capture3.PNG

    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

    RUber

    User Avatar
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted