Calculus III - Interpretations of Partial Derivatives

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Section 13.3 : Interpretations of Partial Derivatives

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1. Determine if $f\left( {x,y} \right) = x\ln \left( {4y} \right) + \sqrt {x + y}$ is increasing or decreasing at $\left( { - 3,6} \right)$ if

we allow $x$ to vary and hold $y$ fixed.
we allow $y$ to vary and hold $x$ fixed.

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a allow

$x$ to vary and hold

$y$ fixed Show Solution

So, we want to determine the increasing/decreasing nature of a function at a point. We know that this means a derivative from our basic Calculus knowledge. Also, from the problem statement we know we want to allow $x$ to vary while $y$ is held fixed. This means that we will need the $x$ partial derivative.

The $x$ partial derivative and its value at the point is,

${f_x}\left( {x,y} \right) = \ln \left( {4y} \right) + \frac{1}{2}{\left( {x + y} \right)^{ - \,\,\frac{1}{2}}}\hspace{0.25in} \to \hspace{0.25in}{f_x}\left( { - 3,6} \right) = \ln \left( {24} \right) + \frac{1}{{2\sqrt 3 }} = 3.4667$

So, we can see that ${f_x}\left( { - 3,6} \right) > 0$ and so at $\left( { - 3,6} \right)$ if we allow $x$ to vary and hold $y$ fixed the function will be increasing.

b allow

$y$ to vary and hold

$x$ fixed Show Solution

This part is pretty much the same as the previous part. The only difference is that here we are allowing $y$ to vary and we’ll hold $x$ fixed. This means we’ll need the $y$ partial derivative.

The $y$ partial derivative and its value at the point is,

${f_y}\left( {x,y} \right) = \frac{x}{y} + \frac{1}{2}{\left( {x + y} \right)^{ - \,\,\frac{1}{2}}}\hspace{0.25in} \to \hspace{0.25in}{f_y}\left( { - 3,6} \right) = - \frac{1}{2} + \frac{1}{{2\sqrt 3 }} = - 0.2113$

So, we can see that ${f_y}\left( { - 3,6} \right) < 0$ and so at $\left( { - 3,6} \right)$ if we allow $y$ to vary and hold $x$ fixed the function will be decreasing.

Note that, because of the three dimensional nature of the graph of this function we can’t expect the increasing/decreasing nature of the function in one direction to be the same as in any other direction!