Interpretation of differentiation results

[musicgold]

Homework Statement

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?

Homework 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.

The Attempt at a Solution

The attached file shows my work.

 

[fourier jr]

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?

 

[musicgold]

not sure what you mean by sensitivity here. is it something to do with the size of the derivative?

Thanks. Yes, by sensitivity I meant how quickly K is changing with a unit change in M.

 

[RUber]

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 $M = -D$.

 

[SammyS]

Homework Statement

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 sensitive 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?

Homework 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.

The Attempt at a Solution

The attached file shows my work.

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.

 

[musicgold]

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 $M = -D$.

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?

 

[RUber]

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.