0
$\begingroup$

Using this post. I am trying to visually and conceptually understand an equation that I have derived at work. My biggest question is essentially from this linked post, but I can't seem to wrap my head around the explanation given there. If I hold one or the other variables constant which of the two has the biggest impact?

The simplified and easiest example of this equation measures the utilization of vehicles by:

$$u = \frac{w}{m^2}$$

utilization = working trucks/ marketable trucks

Taking the gradient as shown in the post linked above, I get:

$$\nabla u(w, m)=\pmatrix{\frac{1}{m^2} \\ -\frac{2w}{m^3}} = \frac{1}{m^2}\pmatrix{1 \\ -\frac{2w}{m}}$$ Using this equation and injecting any point of $w, m$ where $m \le w$ I always get a higher number for $w$, working vehicles because its always positive.

Can someone please help me visualize this?

  • 0
    What do you mean by “largest impact?” If you’re looking for the maximum rate of increase at any point, then examining partial derivatives is the way to go. On the other hand, if you increase both $w$ and $m$ at the same rate, $u$ will eventually get smaller and smaller, so from that perspective it’s $m$ that has the bigger impact on the value. – 2017-02-21
  • 0
    I am looking for the maximum rate of increase at any point. However it seems that $m$ will always be negative? So which direction would I continue to go until I find the point at the relative maximum? – 2017-02-21
  • 0
    The gradient of a function at a point points in the direction of maximum rate of increase, and its length is that rate. – 2017-02-21

0 Answers 0