17. Consider the function f(x, y, z) = xy ln(yz).

Mathematics

1 answer:

Lelu [443]1 year ago

3 0

We can expand

$f(x,y,z) = xy\ln(yz) = xy \ln(y) + xy\ln(z)$

Then using the product and chain rules,

$\dfrac{\partial f}{\partial x} = y\ln(y) + y\ln(z) + \boxed{y\ln(yz)}$

$\dfrac{\partial f}{\partial y} = x\ln(y) + \dfrac{xy}y + x\ln(z) = \boxed{x\ln(yz) + x}$

$\dfrac{\partial f}{\partial z} = \boxed{\dfrac{xy}z}$

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Find a fraction equivalent to 5/7 whose squared terms add up to 1184.

bogdanovich [222]

The system of equations of two unknowns is formulated and solved.

$\large\displaystyle\text{$\begin{gathered}\sf \bf{ \left\{\begin{matrix} \ \ \ \dfrac{x}{y} = \dfrac{5}{7} \\ x^2+y^2 = 1184 \end{matrix}\right. \ \Longrightarrow \ x=\dfrac{5}{7}y } \end{gathered}$}$

$\large\displaystyle\text{$\begin{gathered}\sf \bf{ \left (\dfrac{5}{7}y \right )^2+y^2=1184\ \Longrightarrow\ 25y^2+49y^2=58016 } \end{gathered}$}$

$\large\displaystyle\text{$\begin{gathered}\sf \bf{74y^{2}=58016} \end{gathered}$}\\\large\displaystyle\text{$\begin{gathered}\sf \bf{ \ \ \ \ \ \ y^{2}=784 } \end{gathered}$}\\\large\displaystyle\text{$\begin{gathered}\sf \bf{ \ \ \ \ \ y=\pm\sqrt{784}=\pm28 } \end{gathered}$}$

$\large\displaystyle\text{$\begin{gathered}\sf \bf{ x=\dfrac{5}{7}(\pm 28)=\pm 20 } \end{gathered}$}$

The fraction that satisfies the request is $\bf{\dfrac{20}{28}}$ , since in $\bf{\dfrac{-20}{-28}}$ the negative signs are canceled and the first fraction is obtained.